Mathematical modeling through topological surgery and applications

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we enhance topological surgery with the observed forces and dynamics. We then generalize these low-dimensional cases to a model which extends the formal definition to a continuous process caused by local forces for an arbitrary dimension m. Next, for modeling phenomena which do not happen on arcs, respectively surfaces, but are 2-dimensional, respectively 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further present a dynamical system as a model for both natural phenomena exhibiting a ‘hole drilling’ behavior and our enhanced notion of solid 2-dimensional 0-surgery. Moreover, we analyze the ambient space S in order to introduce the notion of embedded topological surgery in S. This notion is then used for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effects of the process lie beyond the initial manifold, such as the formation of tornadoes. Moreover, we present a visualization of the 4-dimensional process of 3-dimensional surgery by using the new notion of decompactified 2-dimensional surgery and rotations. Finally, we propose a model for a phenomenon exhibiting 3-dimensional surgery: the formation of black holes from cosmic strings. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood.

We are a way for the cosmos to know itself.

Carl Sagan
La réponse est l'homme, quelle que soit la question.

Louis Aragon
There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.

Douglas Adams
Acknowledgments I am grateful to my advisor Sofia Lambropoulou who inspired me to change my career to science. She has supported me throughout the whole duration of my thesis in many ways.
Her guidance and enthusiasm pushed me towards higher goals, taught me how to properly write in a scientific way and opened up new research directions.

Abstract
Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we enhance topological surgery with the observed forces and dynamics. We then generalize these low-dimensional cases to a model which extends the formal definition to a continuous process caused by local forces for an arbitrary dimension m. Next, for modeling phenomena which do not happen on arcs, respectively surfaces, but are 2-dimensional, respectively 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery.
We further present a dynamical system as a model for both natural phenomena exhibiting a 'hole drilling' behavior and our enhanced notion of solid 2-dimensional 0-surgery. Moreover, we analyze the ambient space S 3 in order to introduce the notion of embedded topological surgery in S 3 . This notion is then used for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effects of the process lie beyond the initial manifold, such as the formation of tornadoes. Moreover, we present a visualization of the 4-dimensional process of 3-dimensional surgery by using the new notion of decompactified 2-dimensional surgery and rotations. Finally, we propose a model for a phenomenon exhibiting 3-dimensional surgery: the formation of black holes from cosmic strings. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood. forces acting on a sphere of dimension 0 (that is, two points) or 1 (that is, a circle).
A large part of this work is dedicated to defining new theoretical concepts which are better adapted to the phenomena, to modeling such phenomena in dimensions 1,2 and 3 and to presenting a generalized topological model for m-dimensional n-surgery which captures the observed dynamics. With our enhanced definitions and our model of topological surgery in hand, we match surgery patterns with natural phenomena and we study the physical implications of our modeling. Furthermore, we present a dynamical system that performs a specific type of surgery and we pin down its relation with topological surgery.
Finally, we propose a new type of surgery, the decompactified 2-dimensional surgery, which allows the visualization of 3-dimensional surgery in R 3 . More precisely, the new concepts are: • Continuity and dynamics: In Section 4, we start by enhancing the formal definition of surgery with continuity, whereby an m-dimensional surgery is considered as the continuous local process of passing from an appropriate boundary component of an m + 1-dimensional handle to its complement boundary component. We further notice that surgery in nature is caused by forces. For example, in dimension 1, during meiosis the pairing is caused by mutual attraction of the parts of the chromosomes that are similar or homologous, as detailed and illustrated in Section 5.1. In dimension 2, the creation of tornadoes is caused by attracting forces between the cloud and the earth (as detailed and illustrated in Section 9.3.2), while soap bubble splitting is caused by the surface tension of each bubble which acts as an attracting force (this is discussed in Section 5.4). In Section 5.5 we incorporate these dynamics to our continuous definition and present a model for m-dimensional n-surgery. These dynamics explain the intermediate steps of the formal definition of surgery and extend it to a continuous process caused by local forces. Note that these intermediate steps can also be explained by Morse theory but this approach does not involve the forces.
• Solid surgery: The interior of the initial manifold is now filled in. We observe that phenomena like tension on soap films or the merging of oil slicks are undergoing 1-dimensional surgery but they happen on surfaces instead of 1-manifolds. For example, an oil slick is seen as a disc, which is a continuum of concentric circles together with the center. Similarly, moving up one dimension, during the biological process of mitosis and during tornado formation, 2-dimensional surgery is taking place on 3-dimensional manifolds instead of surfaces. For example, during the process of mitosis, the cell is seen as a 3-ball, that is, a continuum of concentric spheres together with the central point (this is discussed in Section 6.4). Thus, in order to fit natural phenomena where the interior of the initial manifold is filled in, in Section 6, we extend the formal definition by introducing the notion of solid topological surgery in both dimensions 1 and 2.
• Connection with a dynamical system: We establish a connection between these new notions applied on 2-dimensional topological surgery and the dynamical system presented in [30]. In Section 7 we analyze how, with a slight perturbation of parameters, trajectories pass from spherical to toroidal shape through a 'hole drilling' process. We show that our new topological notions are verified by both the local behavior of the steady state points of the system and the numerical simulations of its trajectories. This result gives us on the one hand a mathematical model for 2-dimensional surgery and on the other hand a dynamical system that can model natural phenomena exhibiting this type of surgery.
• Embedded surgery: We notice that in some phenomena exhibiting topological surgery, the ambient space is also involved. For example in dimension 1, during DNA recombination the initial DNA molecule which is recombined can also be knotted. In other words, the initial 1-manifold can be a knot (an embedding of the circle) instead of an abstract circle (see description and illustration in Section 9.1). Similarly in dimension 2, the processes of tornado and black hole formation are not confined to the initial manifold and topological surgery is causing (or is caused by) a change in the whole space (see Section 9.3 and illustrations therein). We therefore define the notion of embedded topological surgery in Section 9 which allows us to model these kind of phenomena but also to view all natural phenomena exhibiting topological surgery as happening in 3-space instead of abstractly.
We consider our ambient 3-space to be S 3 and an extensive analysis of its descriptions together with the presentation of dynamical systems exhibiting its topology is done in Section 8.
• The visualization of 3-dimensional surgery: Finally, in Section 10, we present a way to visualize the 4-dimensional process of 3-dimensional surgery. In order to do so, we introduce the notion of decompactified 2-dimensional surgery which allows us to visualize the process of 2-dimensional surgery in R 2 instead of R 3 . Using this new notion and rotation, we present a way to visualize 3-dimensional surgery in R 3 . This is done in Section 10.2. Further, in Section 10.4, we propose a model for a phenomenon exhibiting 3-dimensional surgery: the formation of black holes from cosmic strings.
This thesis gathers, links and completes the results presented in [18], [19], [3], [2] while extending them one dimension higher. The material is organized as follows: In Section 2 we recall the topological notions that will be used and provide specific examples that will be of great help to readers that are not familiar with the topic. In Section 3, we present the formal definition of topological surgery for an arbitrary dimension m. In Section 4, we enhance the formal definition of surgery with continuity. In Section 5, we introduce dynamics to 1 and 2-dimensional surgery and we discuss natural processes exhibiting these types of surgeries. In Section 5.5, we present a generalized model for m-dimensional n-surgery. In Section 6 we define solid 1 and 2-dimensional surgery and discuss related natural processes. We then present the dynamical system connected to these new notions in Section 7. As all natural phenomena exhibiting surgery (1 or 2-dimensional, solid or usual) take place in the ambient 3-space, in Section 8 we present the 3-sphere S 3 and the duality of its descriptions. This allows us to define in Section 9 the notion of embedded surgery. Finally, in Section 10, we use lower dimensional surgeries to visualize 3-dimensional surgery and propose a topological model for black hole formation from cosmic strings.

Useful mathematical notions
In this section we introduce basic notions related to topological surgery. Reader that are familiar with the formalism of the topic can directly move to the formal definition in Section 3.

Manifolds
An n-manifold without boundary is a 'nice' topological space with the property that each point in it has a neighborhood topologically equivalent to the usual ndimensional Euclidean space R n . In other words an n-manifold resembles locally Similarly, an n-manifold with boundary is 'nice' topological space with the property that each point in it has a neighborhood topologically equivalent either to R n (if the point lies in the interior) or R n + (if the point lies on the boundary).

Homeomorphisms
In Section 2.1 by 'topologically equivalent' we mean the following: two n-manifolds X and Y are homeomorphic or topologically equivalent if there exists a homeomorphism between them, namely a function f : X → Y with the properties that: f is continuous There exists the inverse function f −1 : Y → X (equivalently f is 1-1 and onto) f −1 is also continuous Intuitively the homeomorphism f is an elastic deformation of the space X to the space Y , not involving any self-intersections or any 'cutting' and 'regluing' (see also Appendix A).

Properties of manifolds
An n-manifold, M , is said to be: connected if it consists of only one piece, compact if it can be enclosed in some k-dimensional ball, orientable if any oriented frame that moves along any closed path in M returns to a position that can be transformed to the initial one by a rotation.
The above notions are more rigorously defined in Appendix A.

n-spheres and n-balls
In each dimension the basic connected, oriented, compact n-manifold without boundary is the n-sphere, S n . Also, the basic connected, oriented n-manifold with boundary is the n-ball, D n . The boundary of a n-dimensional ball is a n − 1−dimensional sphere, ∂D n = S n−1 , n >= 1. In Fig 1, this relation is shown for n = 1, 2 and 3. As shown in Fig 1 (1), the space S 0 is the disjoint union of two points. By convention, we consider these two one-point spaces to be {+1} and {−1}: S 0 = {+1} {−1}. Note that, in this analysis, we will follow this convention by considering that n-spheres and n-balls are centered at the origin.
Besides the relation of S n with D n+1 described above, the n-sphere S n is also intrinsically related to the Euclidian space R n via the notion of compactification.

The compactification of R n
Compactification is the process of making a topological space into a compact space. For each dimension n, the space R n with all points at infinity compactified to one single point is homeomorphic to S n . So, S n is also called the one-point compactification of R n . Conversely, a sphere S n can be decompactified to the space R n by the so-called

Product spaces
The product space of two manifolds X and Y is the manifold made from their Cartesian product X × Y (see also Appendix A). If X, Y are manifolds with boundary, the boundary of product space X × Y is: For example the next common connected, oriented, compact 2-manifold without boundary after S 2 is the torus, which can be perceived as the boundary of a doughnut, and it is the product space S 1 × S 1 . Analogously, a solid torus, which can be perceived as a whole doughnut, is the product space S 1 × D 2 . A solid torus is a 3-manifold with boundary a torus: Other product spaces that we will be using here are: the cylinder S 1 × D 1 or D 1 × S 1 (see Fig 3), the solid cylinder D 2 × D 1 which is homeomorphic to the 3-ball and the spaces of the type S 0 × D n , which are the disjoint unions of two n-balls D n D n .

Figure 3. Two ways of viewing a cylinder
All the above examples of product spaces that are of the form S p × D q can be viewed as q-thickenings of the p-sphere. For example the 2-thickening of S 0 comprises two discs, while the 3-thickening of S 0 comprises two 3-balls. It is also worth noting that the product spaces S p × D q and D p+1 × S q−1 have the same boundary:

Embeddings
An embedding of an n-manifold N n in an m-manifold M m is a map f : N → M such that its restriction on the image f (N ) is a homeomorphism between N and f (N ).
The notion of embedding allows to view spaces inside specific manifolds instead of abstractly. Embeddings even of simple manifolds can be very complex. For example, the embeddings of the circle S 1 in the 3-space R 3 are the well-known knots whose topological classification is still an open problem of low-dimensional topology.
An embedding of a submanifold N n → M m is framed if it extends to an embedding A framed n-embedding in M is an embedding of the (m − n)-thickening of the n-sphere, h : S n × D m−n → M , with core n-embedding e = h | : S n = S n × {0} → M .
For example, the framed 1-embeddings in R 3 comprise embedded solid tori in the 3-space with core 1-embeddings being knots.
Let X, Y be two n-manifolds with homeomorphic boundaries ∂X and ∂Y (which are (n − 1)−manifolds). Let also h denote a homeomorphism h : ∂X → ∂Y . Then, from X ∪ Y one can create a new n-manifold without boundary by 'gluing' X and Y along their boundaries. The gluing is realized by identifying each point x ∈ ∂X to the point h(x) ∈ ∂Y . The map h is called gluing homeomorphsim, see Appendix A.
One important example is the gluing of two n-discs along their common boundary which gives rise to the n-sphere, see Fig 4 for n = 1, 2. For n = 3, the gluing of two 3-balls yielding the 3-sphere S 3 is illustrated and discussed in Section 8.1.2. Another interesting example is the gluing of solid tori which also yield the 3-sphere. This is illustrated and discussed in Section 8.1.3 As we will see in next section, the notions of embedding and gluing homeomorphism together with property ( ) described in 2.6 are the key ingredients needed to define topological surgery. It is roughly the procedure of removing an embedding of S p × D q and gluing back D p+1 × S q−1 along their common boundary.

The formal definition of surgery
We recall the following well-known definition of surgery: Definition 1. An m-dimensional n-surgery is the topological procedure of creating a new m-manifold M out of a given m-manifold M by removing a framed n-embedding h : S n ×D m−n → M , and replacing it with D n+1 ×S m−n−1 , using the 'gluing' homeomorphism h along the common boundary S n × S m−n−1 . Namely, and denoting surgery by χ : The symbol 'χ' of surgery comes from the Greek word 'χ ιρoυργική' (cheirourgiki) whose term 'cheir' means hand. Note that from the definition, we must have n + 1 ≤ m. Also, the horizontal bar in the above formula indicates the topological closure of the set underneath.
Further, the dual m-dimensional (m − n − 1)-surgery on M removes a dual framed and replaces it with S n × D m−n , using the 'gluing' homeomorphism g (or h −1 ) along the common boundary S n × S m−n−1 . That is: The resulting manifold χ(M ) may or may not be homeomorphic to M . From the above definition, it follows that M = χ −1 (χ(M )). Preliminary definitions behind the definition of surgery such as topological spaces, homeomorphisms, embeddings and other related notions are provided in Section 2 and Appendix A. For further reading, excellent references on the subject are [26,28,29]. We shall now apply the above definition to dimensions 1 and 2.

1-dimensional 0-surgery
We only have one kind of surgery on a 1-manifold M , the 1-dimensional 0-surgery where m = 1 and n = 0: The above definition means that two segments S 0 × D 1 are removed from M and they are replaced by two different segments D 1 × S 0 by reconnecting the four boundary points S 0 × S 0 in a different way. In Fig. 5 (a) and 6 (a), S 0 × S 0 = {1, 2, 3, 4}. As one possibility, if we start with M = S 1 and use as h the standard (identity) embedding denoted with h s , we obtain two circles S 1 × S 0 . Namely, denoting by 1 the identity homeomorphism, we Fig. 5 (a). However, we can also obtain one circle S 1 if h is an embedding h t that reverses the orientation of one of the two arcs of S 0 × D 1 . Then in the substitution, joining endpoints 1 to 3 and 2 to 4, the two new arcs undergo a half-twist, see Fig. 6 (a). More specifically, if we take D 1 = [−1, +1] and define the homeomorphism ω : D 1 → D 1 ; t → −t, the embedding used in Fig. 6 (a) is between the embeddings h s and h t of S 0 × D 1 can be clearly seen by comparing the four boundary points 1, 2, 3 and 4 in Fig. 5 (a) and Fig. 6 (a). Note that in dimension one, the dual case is also an 1-dimensional 0-surgery. For

2-dimensional 0-surgery
Starting with a 2-manifold M , there are two types of surgery. One type is the 2dimensional 0-surgery, whereby two discs S 0 × D 2 are removed from M and are replaced in the closure of the remaining manifold by a cylinder D 1 × S 1 , which gets attached via a homeomorphism along the common boundary S 0 × S 1 comprising two copies of S 1 . The gluing homeomorphism of the common boundary may twist one or both copies of S 1 . For M = S 2 the above operation changes its homeomorphism type from the 2-sphere to that of the torus. View Fig. 5 (b 1 ) for the standard embedding h s and Fig. 6 (b 1 ) for a twisting embedding h t . For example, the homeomorphism µ : induces the 2-dimensional analogue h t of the embedding defined in the previous example, now, the cylinder D 1 × S 1 is glued along the common boundary S 0 × S 1 , the twisting of this boundary induces the twisting of the cylinder, see Fig. 6 (b 1 ).

2-dimensional 1-surgery
The other possibility of 2-dimensional surgery on M is the 2-dimensional 1-surgery: here a cylinder (or annulus) S 1 × D 1 is removed from M and is replaced in the closure of the remaining manifold by two discs D 2 × S 0 attached along the common boundary S 1 × S 0 .
For M = S 2 the result is two copies of S 2 , see This operation corresponds to fixing the circle S 1 bounding the right side of the cylinder S 1 × D 1 , rotating the circle Next, for m = 3 and n = 2, we have the 3-dimensional 2-surgery but we will not analyze this type of surgery as it is the reverse process of 3-dimensional 0-surgery. Finally, for m = 3 and n = 1, we have the 3-dimensional 1-surgery whereby a solid torus S 1 × D 2 is removed from M and is replaced by another solid torus D 2 × S 1 : Both processes will be analyzed and visualized in Section 10.

Continuity
As we will see in the following sections, topological surgery happens in nature as a continuous process caused by local forces. However, the formal definition of surgery

The continuous definition of surgery
Let us first notice that if we glue together the two m-manifolds involved in the process of m-dimensional n-surgery along their common boundary we obtain the m-sphere. Namely, from Definition 1 and using property ( ) discussed in Section 2.6, we can see that  . This process together with continuous 1-dimensional surgery will be elaborated in much more details in the next section where dynamics are also introduced.

Dynamics
In this section we present natural processes exhibiting topological surgery in dimensions 1 and 2 and we incorporate the observed dynamics to Definition 2, thus creating a model which extends surgery to a continuous process caused by local forces. Further, for each dimension, we go back to the phenomena and pin down the forces introduced by our models.

Dynamic 1-dimensional topological surgery
We find that 1-dimensional 0-surgery is present in phenomena where 1-dimensional splicing and reconnection occurs. It can be seen for example during meiosis when new combinations of genes are produced, see Fig. 8 (3), during magnetic reconnection, the phenomena whereby cosmic magnetic field lines from different magnetic domains are spliced to one another, changing their pattern of conductivity with respect to the sources, see Fig. 8 (2) from [5] and in site-specific DNA recombination (see Fig 22) whereby nature alters the genetic code of an organism, either by moving a block of DNA to another position on the molecule or by integrating a block of alien DNA into a host genome (see [32]). It is worth mentioning that 1-dimensional 0-surgery is also present during the reconnection of vortex tubes in a viscous fluid and quantized vortex tubes in superfluid helium. As mentioned in [17], these cases have some common qualitative features with the magnetic reconnection shown in Fig. 8 (2).
Since all the above phenomena follow a dynamic process, in Fig. 8 (1), we introduce dynamics to Definition 2 which shows how the intermediate steps of surgery are caused by local forces. The process starts with the two points (in red) specified on any 1-dimensional manifold, on which attracting forces are applied (in blue). We assume that these forces are caused by an attracting center (also in blue). Then, the two segments S 0 × D 1 , which are neighborhoods of the two points, get close to one another. When the specified points (or centers) of the two segments reach the attracting center they touch and recoupling takes place, giving rise to the two final segments D 1 × S 0 , which split apart. In Fig. 8 (1), case (s) corresponds to the identity embedding h s described in Section 3.1, while (t) corresponds to the twisting embedding h t described in Section 3.1. As also mentioned in Section 3.1, the dual case is also a 1-dimensional 0-surgery, as it removes segments D 1 × S 0 and replaces them by segments S 0 × D 1 . This is the reverse process which starts from the end and is illustrated in Fig. 8 (1) as a result of the orange forces and attracting center which are applied on the 'complementary' points.
Note that these local dynamics produce different manifolds depending on where the initial neighborhoods are embedded. Taking the known case of the standard embedding h s and M = S 1 , we obtain S 1 × S 0 , see Fig 9 (a). Furthermore, as shown in Fig 9 (b), we also obtain S 1 × S 0 even if the attracting center is outside S 1 . Note that these outcomes are not different than the ones shown in formal surgery (recall   (1) where the two segments get close to one another. For t = 0 it is the two straight lines where the reconnection takes place as shown in the third stage of Fig. 8 (1) while for t = 1 it represents the hyperbola of the two final segments shown in case (s) of the fourth stage of Fig. 8 (1). This sequence can be generalized for higher dimensional surgeries as well.
For example, for 2-dimensional surgery, we can visualize the process by varying parameter t of equation x 2 + y 2 − z 2 = t. We mention this approach because it gives us an algebraic formulation of surgery's time evolution. However, in this analysis we will not use it as we are focusing on the introduction of forces and the attracting center.

Modeling phenomena exhibiting 1-dimensional surgery
The aforementioned phenomena are all 1-dimensional in the sense that they involve 1dimensional surgery happenning in a initial 1-dimensional manifold. We will take a closer look at them and show that the described dynamics and attracting forces introduced by our model are present in all cases. Namely, magnetic reconnection ( Fig. 8 (2)) corresponds to a dual 1-dimensional 0-surgery (see case (t) of Fig. 8 (1)) where g : D 1 × S 0 → M is a dual embedding of the twisting homeomorphism h t defined in Section 3.1. The tubes are viewed as segments and correspond to an initial manifold M = S 0 × D 1 (or M = S 1 if they are connected) on which the local dynamics act on two smaller segments S 0 × D 1 .
Namely, the two magnetic flux tubes have a nonzero parallel net current through them, which leads to attraction of the tubes (cf. [16]). Between them, a localized diffusion region develops where magnetic field lines may decouple. Reconnection is accompanied with a sudden release of energy and the magnetic field lines break and rejoin in a lower energy state.
In the case of chromosomal crossover during meiosis ( Fig. 8 (3)), we have the same dual 1-dimensional 0-surgery as magnetic reconnection (see case (t) of Fig. 8 (1)). During this process, the homologous (maternal and paternal) chromosomes come together and pair, or synapse, during prophase. The pairing is remarkably precise and is caused by mutual attraction of the parts of the chromosomes that are similar or homologous. Further, each paired chromosomes divide into two chromatids. The point where two homologous non-sister chromatids touch and exchange genetic material is called chiasma. At each chiasma, two of the chromatids have become broken and then rejoined (cf. [27]). In this process, we consider the initial manifold to be one chromatid from each chromosome, hence the initial manifold is M = S 0 × D 1 on which the local dynamics act on two smaller For site-specific DNA recombination (see Fig 22), we have a 1-dimensional 0surgery (see case (t) of Fig. 8 (1)). Here the initial manifold is a knot which is an embedding of M = S 1 in 3-space but this will be detailed in Section 9.1. As mentioned in [11], enzymes break and rejoin the DNA strands, hence in this case the seeming attraction of the two specified points is realized by the enzyme. Note that, while both are genetic recombinations, there is a difference between chromosomal crossover and site-specific DNA recombination. Namely, chromosomal crossover involves the homologous recombination between two similar or identical molecules of DNA and we view the process at the chromosome level regardless of the knotting of DNA molecules.
Finally, vortices reconnect following the steps of 1-dimensional 0-surgery with a standard embedding shown in (see case (s) of Fig. 8 (1)). The initial manifold is again As mentioned in [13], the interaction of the anti-parallel vortices goes from attraction before reconnection, to repulsion after reconnection. stabilized contra-rotating identations in the water-air discontinuity surface of a swimming pool, see [14] for details). It can also be seen in gene transfer in bacteria where the donor cell produces a connecting tube called a 'pilus' which attaches to the recipient cell, see  In Fig. 10 (1), below the instances of the standard embedding, we also show the instances of these processes when a non-trivial embedding are used (recall Section 3.2).

Dynamic 2-dimensional topological surgery
Note that these embeddings are more appropriate for natural processes involving twisting, such as tornadoes and Falaco solitons. In this example of twisted 2-dimensional 0-surgery, the two discs S 0 × D 2 are embedded via a twisted homemorphism h t while, in the dual case, the cylinder D 1 × S 1 is embedded via a twisted homemorphism g t . Here h t rotates the two initial discs in opposite directions by an angle of 3π/4 and we can see how this rotation induces the twisting of angle 3π/2 of the final cylinder (which corresponds to homemorphism g t rotating the top and bottom of the cylinder by 3π/4 and −3π/4 respectively). More specifically, if we define the homeomorphism ω 1 , ω 2 : to be rotations by 3π/4 and −3π/4 respectively, then h t is defined as the composition analogously.
The process of dynamic 2-dimensional 0-surgery starts with two points, or poles, specified on the manifold (in red) on which attracting forces caused by an attracting center are applied (in blue). Then, the two discs S 0 × D 2 , neighbourhoods of the two poles, approach each other. When the centers of the two discs touch, recoupling takes place and the discs get transformed into the final cylinder D 1 × S 1 , see Fig. 10 (1). The cylinder created during 2-dimensional 0-surgery can take various forms. For example, it is a tubular vortex of air in the case of tornadoes, a transverse torsional wave in the case of Falaco solitons and a pilus joining the genes in gene transfer in bacteria.
On the other hand, phenomena exhibiting 2-dimensional 1-surgery are the result of an infinitum of coplanar attracting forces which 'collapse' a cylinder, see Fig. 10 (1) from the end. As mentioned in Section 3.3, the dual case of 2-dimensional 0-surgery is the 2-dimensional 1-surgery and vice versa. This is illustrated in Fig. 10 (1) where the reverse process is the 2-dimensional 1-surgery which starts with the cylinder and a specified circular region (in red) on which attracting forces caused by an attracting center are applied (in orange). A 'necking' occurs in the middle which degenerates into a point and finally tears apart creating two discs S 0 × D 2 . This cylinder can be embedded, for example, in the region of the bubble's surface where splitting occurs, on the region of metal specimens where necking and fracture occurs or on the equator of the cell which is about to undergo a mitotic process. In Fig 11 (a) and (b), we apply the local dynamics of Fig. 10 (1) to the initial manifold M = S 2 and produce the same manifolds seen in formal 2-dimensional surgery (recall ) through a continuous process resulting of forces. Note that, as also seen in 1-dimensional surgery (Fig 9 (b)), if the blue attracting center in Fig 11 (a) was outside the sphere and the cylinder was attached on S 2 externally, the result would still be a torus.
Finally, it is worth pointing out that these local dynamics produce different manifolds depending on the initial manifold where they act. Taking examples from natural phenomena, 2-dimensional 0-surgery transforms an M = S 0 × S 2 to an S 2 by adding a cylinder during gene transfer in bacteria (see Fig. 10 (4)) but can also transform an M = S 2 to a torus by 'drilling out' a cylinder during the formation of Falaco solitons (see Fig. 10 (3)) in which case S 2 is the pool of water and the cylinder is the boundary of the tubular neighborhood around the thread joining the two poles.
Remark 3. Note that Remark 1 is also true here. One can obtain Fig. 10 (1) by rotating Fig. 8 (1) and this extends also to the dynamics and forces. For instance, by rotating the two points, or S 0 , on which the pair of forces of 1-dimensional 0-surgery acts (shown in red in the last instance of Fig. 8 (1)) by 180°around a vertical axis we get the circle, or S 1 , on which the infinitum of coplanar attracting forces of 2-dimensional 1-surgery acts (shown in red in the last instance of Fig. 10 (1)).

Modeling phenomena exhibiting 2-dimensional surgery
Looking back at the natural phenomema happening on surfaces, an example is soap bubble splitting during which a soap bubble splits into two smaller bubbles. This process is the 2-dimensional 1-surgery on M = S 2 shown in Fig 11 (b). The orange attracting force in this case is the surface tension of each bubble that pulls molecules into the tightest possible groupings.
If one looks closer at the other phenomena exhibiting 2-dimensional surgery shown in Fig. 10, one can see that these phenomena do not happen on surfaces but on 3-dimensional manifolds, therefore we can't model them as 2-dimensional surgeries. As we will see in Section 6, these processes are described by the notion of solid surgery. Therefore they will be analyzed after the introduction of this notion. For instance, gene transfer in bacteria, drop coalescence and the formation of Falaco solitons are discussed in Section 6.3 while mitosis and fracture will be discussed in in Section 6.4.
Moreover, as we will see in Section 9, the ambient space is also involved in the process of tornado formation, see Fig. 10 (2). Therefore it will analyzed in Section 9.3.2, after the introduction of the notion of embedded surgery.

A model for dynamic m-dimensional n-surgery
As mentioned in Section 4.1, surgery can be viewed as collapsing the thickened core S n to a singular point and then uncollapsing the thickened core S m−n−1 . As seen in Fig. 10 (1), in the case of 2-dimensional 0-surgery, forces (in blue) are applied to core S 0 , whose thickening comprises the two discs, while in the case of the 2-dimensional 1-surgery, forces (in orange) are applied on the core S 1 , whose thickening is the cylinder. In other words, the forces that model 2-dimensional n-surgery are always applied to the core n-embedding This observation can be generalized as follows: The process followed by natural phenomena exhibiting topological surgery is modeled by Definition 2 enhanced with attracting forces acting on the cores S n and S m−n−1 of embeddings S n × D m−n and D n+1 × S m−n−1 . Moreover, we view this continuous passage as a result of forces towards the attracting center, which is identified with the singular point.
The above are shown in Fig. 12 (1) and (2)   (1) Model for dynamic 1-dimensional 0-surgery (2) Model for dynamic 2-dimensional 0-surgery process is that a solid cylindrical region located in the center of the cell collapses and a D 3 is transformed into an S 0 × D 3 . Similarly, during tornado formation, the created cylinder is not just a cylindrical surface D 1 × S 1 but a solid cylinder D 2 × S 1 containing many layers of air (this phenomena will be detailed in Section 9.3.2). Of course we can say that, for phenomena involving 3-dimensional manifolds, the outer layer of the initial manifold is undergoing 2-dimensional surgery. In this section we will define topologically what happens to the whole manifold.
The need of such a definition is also present in dimension 1 for modeling phenomena such as the merging of oil slicks and tension on membranes (or soap films). These phenomena undergo the process of 1-dimensional 0-surgery but happen on surfaces instead of 1-manifolds.
We will now introduce the notion of solid surgery (in both dimensions 1 and 2) where the interior of the initial manifold is filled in. There is one key difference compared to the dynamic surgeries discussed in the previous section. While the local dynamics described in Fig. 8 and 10 can be embedded in any manifold, here we also have to fix the initial manifold in order to define solid surgery. For example, as we will see next, we define separately the processes of solid 1-dimensional 0-surgery on D 2 and solid 1-dimensional 0-surgery on D 2 × S 0 . However, the underlying features are common in both.

Solid 1-dimensional topological surgery
Solid 1-dimensional 0-surgery on the 2-disc D 2 is the topological procedure whereby a ribbon D 1 × D 1 is being removed, such that the closure of the remaining manifold comprises two discs D 2 × S 0 . The reader is referred to Fig 5 (a) where the interior is now supposed to be filled in. This process is equivalent to performing 1-dimensional 0-surgeries on the whole continuum of concentric circles included in D 2 , see Fig. 13. More precisely, and introducing at the same time dynamics, we define: Definition 3. Solid 1-dimensional 0-surgery on D 2 is the following process. We start with the 2-disc of radius 1 with polar layering: where r the radius of a circle and P the limit point of the circles, which is the center of the disc and also the circle of radius zero. We specify colinear pairs of antipodal points, all on the same diameter, with neighbourhoods of analogous lengths, on which the same colinear attracting forces act. See Fig 13 (1) where these forces and the attracting center are shown in blue. Then, in (2), the antipodal segments get closer to one another or, equivalently, closer to the attracting center. Note that here, the attracting center coincides with the limit point of all concentric circles, which is shown in green from instance (2) and on. Then, as shown from (3) to (9), we perform 1-dimensional 0-surgery on the whole continuum of concentric circles. The natural order of surgeries is as follows: first, the center of the segments that are closer to the center of attraction touch, see (4). After all other points have also reached the center, see (5), decoupling starts from the central or limit point. We define 1-dimensional 0-surgery on the limit point P to be the two limit points of the resulting surgeries. That is, the effect of solid 1-dimensional 0-surgery on a point is the creation of two new points, see (6). Next, the other segments reconnect, from the inner, see (7), to the outer ones, see (8), until we have two copies of D 2 , see (9) and (10). The proposed order of reconnection, from inner to outer, is the same as the one followed by skin healing, namely, the regeneration of the epidermis starts with the deepest part and then migrates upwards. The above process is the same as first removing the center P from D 2 , doing the 1-dimensional 0-surgeries and then taking the closure of the resulting space. The resulting manifold is which comprises two copies of D 2 .
We also have the reverse process of the above, namely, solid 1-dimensional 0-surgery on two discs D 2 × S 0 is the topological procedure whereby a ribbon D 1 × D 1 joining the discs is added, such that the closure of the remaining manifold comprise one disc D 2 .
This process is the result of the orange forces and attracting center which are applied on the 'complementary' points, see Fig. 13 in reverse order. This operation is equivalent to performing 1-dimensional 0-surgery on the whole continuum of pairs of concentric circles in D 2 D 2 . We only need to define solid 1-dimensional 0-surgery on two limit points to be the limit point P of the resulting surgeries. That is, the effect of solid 1-dimensional 0-surgery on two points is their merging into one point. The above process is the same as first removing the centers from the D 2 × S 0 , doing the 1-dimensional 0-surgeries and then taking the closure of the resulting space. The resulting manifold is which comprises one copy of D 2 .

Solid 2-dimensional topological surgery
Moving up one dimension, there are two types of solid 2-dimensional surgery on the 3-ball, , which is two copies of D 3 . See Fig 5 (b 2 ) where the interior is supposed to be filled in. Those processes are equivalent to performing 2-dimensional surgeries on the whole continuum of concentric spheres included in D 3 . More precisely we have: Definition 4. We start with the 3-ball of radius 1 with polar layering: where r the radius of the 2-sphere S 2 r and P the limit point of the spheres, that is, their common center and the center of the ball. The process is characterized on one hand by the 1-dimensional core L of the solid cylinder which joins the two selected antipodal points of the outer shell and intersects each spherical layer at its two corresponding antipodal points, and on the other hand by the embedding h. The process results in a continuum of layered tori and can be viewed as drilling out a tunnel along L according to h. Note that in Fig. 14, the identity embedding has been used. However, a twisting embedding, which is the case shown in Fig. 10 (1), agrees with our intuition that, for opening a hole, drilling with twisting seems to be the easiest way. Examples of these two embeddings can be found in Section 3.2. For both types of solid 2-dimensional surgery, the above process is the same as: first removing the center P from D 3 , performing the 2-dimensional surgeries and then taking the closure of the resulting space. Namely we obtain: which is a solid torus in the case of solid 2-dimensional 0-surgery and two copies of D 3 in the case of solid 2-dimensional 1-surgery.
As seen in Fig. 14, we also have the two dual solid 2-dimensional surgeries, which represent the reverse processes. As already mentioned in Section 3.3, the dual case of 2-dimensional 0-surgery is the 2-dimensional 1-surgery and vice versa. More precisely: Definition 5. The dual case of solid 2-dimensional 0-surgery on D 3 is the solid 2dimensional 1-surgery on a solid torus D 2 × S 1 whereby a solid cylinder D 1 × D 2 filling the hole is added, such that the closure of the resulting manifold comprises one 3-ball D 3 . This is the reverse process shown in Fig. 14 (b 1 ) which results from the orange forces and attracting center. Given that the solid torus can be written as a union of nested tori together with the core circle: 1-surgeries are performed on each toroidal layer starting from specified annular peels of analogous sizes where the same coplanar forces act on the central rings of the annuli. These forces are caused by the same attracting center lying outside the torus. It only remain to define the solid 2-dimensional 1-surgery on the limit circle to be the limit point P of the resulting surgeries. That is, the effect of solid 2-dimensional 1-surgery on the core circle is that it collapses into one point, the attracting center. The above process is the same as first removing the core circle from D 2 × S 1 , doing the 2-dimensional 1-surgeries on the layered tori, with the same coplanar acting forces, and then taking the closure of the resulting space. Hence, the resulting manifold is which comprises one copy of D 3 .
Further, the dual case of solid 2-dimensional 1-surgery on D 3 is the solid 2-dimensional 0-surgery on two 3-balls D 3 whereby a solid cylinder D 2 × D 1 joining the balls is added, such that the closure of the resulting manifold comprise of one 3-ball D 3 . This is the reverse process shown in Fig. 14 (b 2 ) which results from the blue forces and attracting center. We only need to define the solid 2-dimensional 0-surgery on two limit points to be the limit point P of the resulting surgeries. That is, as in solid 1-dimensional surgery (see which comprises one copy of D 3 . Note that remarks 1 and 3 are also true here. One can obtain the instances of solid 2-dimensional surgeries ( Fig. 14 (b 1 ) and Fig. 14 (b 2 )) by rotating the instances of solid 1-dimensional surgery ( Fig. 14 (a)) respectively by 180°around a vertical axis and by 180°a round a horizontal axis.

Modeling phenomena exhibiting solid 2-dimensional 0-surgery
We will now describe the phenomena mentioned in Section 5.4 which undergo the process of solid 2-dimensional 0-surgery.
For gene transfer in bacteria, see Fig. 10 (4) (also, for description and instructive illustrations see [8]), the donor cell produces a connecting tube called a 'pilus' which attaches to the recipient cell, brings the two cells together and transfers the donor's DNA.
This process is similar to the one shown earlier in Fig 14 (b 2 ) as two copies of D 3 merge into one, but here the attracting center is located on the recipient cell. This process is a Similarly, the process of drop coalescence is also a solid 2-dimensional 0-surgery on two 3-balls M = D 3 × S 0 , see Fig 14 (b 2 ). The process of drop coalescence also exhibits the forces of our model. Namely, the surfaces of two drops must be in contact for coalescence to occur. This surface contact is dependent on both the van der Waals attraction and the surface repulsion forces between two drops. When the van der Waals forces cause rupture of the film, the two surface films are able to fuse together, an event more likely to occur in areas where the surface film is weak. The liquid inside each drop is now in direct contact, and the two drops are able to merge into one.
Moreover, as already mentioned in Section 5.3, the formation of Falaco solitons is a natural phenomenon exhibiting solid 2-dimensional 0-surgery, see Fig. 10 (3) (for photos of pairs of Falaco solitons in a swimming pool, see [14]). Note that the term 'Falaco Soliton' appears in 2001 in [15]. These pairs of singular surfaces (poles) are connected by means of a stabilizing thread. The two poles get connected and their rotation propagates below the water surface along the joining thread and the tubular neighborhood around it. This process is a solid 2-dimensional 0-surgery with a twisted homeomorphism (see Fig. 10 (1)) where the initial manifold is the water contained in the volume of the pool where the process happens, which is homeomorphic to a 3-ball, that is M = D 3 . It is also worth mentioning that the creation of Falaco solitons is immediate and does not allow us to see whether the transitions of solid 2-dimensional 0-surgery shown in Fig. 14 (b 1 ) are followed or not. However, these dynamics are certainly visible during the annihilation of Falaco solitons. Namely, when the topological thread joining the poles is cut, the tube tears apart and slowly degenerates to the poles until they both stops spinning and vanish. Therefore, the continuity of our dynamic model is clearly present during the reverse process which corresponds to a solid 2-dimensional 1-surgery on a pair of Falaco solitons, that is, a solid torus D 2 × S 1 degenerating into a still swimming pool D 3 , see the reverse process of Fig. 14 (b 1 ).
Note that it is conjectured in [14] that the coherent topological features of the Falaco solitons and, by extension, the process of solid 2-dimensional 0-surgery appear in both macroscopic level (for example in the Wheeler's wormholes) and microscopic level (for example in the spin pairing mechanism in the microscopic Fermi surface). For more details see [14].

Modeling phenomena exhibiting solid 2-dimensional 1-surgery
We will now describe the phenomena mentioned in Section 5.4 which undergo the process of solid 2-dimensional 1-surgery.
As already mentioned, the collapsing of the central disc of the sphere caused by the orange attracting forces in Fig. 14 (b 2 ) can also be caused by pulling apart left and right hemispheres of the 3-ball D 3 , that is, the causal forces can also be repelling. For example, during the fracture of metal specimens under tensile forces, solid 2-dimensional 1-surgery is caused by forces that pull apart each end of the specimen. On the other hand, in the biological process of mitosis, both attracting and repelling forces forces are present. We will now describe these two processes in details.
When the tension applied on metal specimens by tensile forces results in necking  (7) (for description and instructive illustrations see for example [12]). We will see that both aforementioned forces are present here. During mitosis, the chromosomes, which have already duplicated, condense and attach to fibers that pull one copy of each chromosome to opposite sides of the cell (this pulling is equivalent to repelling forces). The cell pinches in the middle and then divides by cytokinesis. The structure that accomplishes cytokinesis is the contractile ring, a dynamic assembly of filaments and proteins which assembles just beneath the plasma membrane and contracts to constrict the cell into two (this contraction is equivalent to attracting forces). In the end, two genetically-identical daughter cells are produced. we connect topological surgery, enhanced with these notions, with a dynamical system.
We will see that, with a small change in parameters, the trajectories of its solutions are performing solid 2-dimensional 0-surgery. Therefore, this dynamical system constitutes a specific set of equations modeling natural phenomena undergoing solid 2-dimensional 0-surgery. More specifically, we will see that the change of parameters of the system affects the eigenvectors and induces a flow along a segment joining two steady state points. The induced flow represents the attracting forces shown in Fig. 14 (b 1 ). Finally, we will see how our topological definition of solid 2-dimensional 0-surgery presented in Section 6.2 is verified by our numerical simulations and, in particular, that surgery on a steady point creates a limit cycle.

The dynamical system and its steady state points
In [30], N.Samardzija and L.Greller study the behavior of the following dynamical system (Σ) that generalizes the classical Lotka-Volterra problem [21,33] into three dimensions: In subsequent work [31], the authors present a slightly different model, provide additional numerical simulations and deepen the qualitative analysis done in [30]. Since both models coincide in the parametric region we are interested in, we will use the original model and notation and will briefly present some key features of the analyses done in [30] and [31].
The system (Σ) is a two-predator and one-prey model, where the predators Y, Z do not interact directly with one another but compete for prey X. As X, Y, Z are populations, only the positive solutions are considered in this analysis. It is worth mentioning that, apart from a population model, (Σ) may also serve as a biological model and a chemical model, for more details see [30].
The parameters A, B, C are analyzed in order to determine the bifurcation properties of the system, that is, to study the changes in the qualitative or topological structure of the family of differential equations (Σ). As parameters A, B, C affect the dynamics of constituents X, Y, Z, the authors were able to determine conditions for which the ecosystem of the three species results in steady, periodic or chaotic behavior. More precisely, the authors derive five steady state solutions for the system but only the three positive ones are taken into consideration. These points are: It is worth reminding here that a steady state (or singular) point of a dynamical system is a solution that does not change with time.

Local behavior and numerical simulations
Let, now, J(S i ) be the Jacobian of (Σ) evaluated at S i for i = 1, 2, 3 and let the sets Γ{J(S i )} and W {J(S i )} to be, respectively, the eigenvalues and the corresponding associated eigenvectors of J(S i ). These are as follows: Using the sets of eigenvalues and eigenvectors presented above, the authors characterize in [30], [31] the local behavior of the dynamical system around these three points using • Region (a) Setting B/A = 1 and equating the right side of (Σ) to zero, one finds as solution the one-dimensional singular manifold: that passes through the points S 2 and S 3 . Since all points on L are steady state points, there is no motion along it. For (1/8B−1) A/B < C ≤ 2(1+ √ 2), S 2 is an unstable center while S 3 is a stable center (for a complete analysis of all parametric regions see [30]). This means that if λ 1 , λ 2 , λ 3 denote the eigenvalues of either S 2 or S 3 with λ 1 ∈ R and λ 2 , λ 3 ∈ C, For B/A > 1 and (1/8B − 1) A/B < C ≤ 2(1 + √ 2), S 2 is an inward unstable vortex and S 3 is an outward stable vortex. This means that in both cases they must satisfy the conditions λ 1 ∈ R and λ 2 , λ 3 ∈ C with λ 3 = λ * 2 , the conjugate of λ 2 . The eigenvalues of S 2 must further satisfy λ 1 < 0 and Re(λ 2 ) = Re(λ 3 ) > 0, while the eigenvalues of S 3 must further satisfy λ 1 > 0 and Re(λ 2 ) = Re(λ 3 ) < 0. The local behaviors around S 2 and S 3 for this parametric region are shown in Fig 15 (b). It is worth mentioning that Fig 15 (b) reproduces Fig 1 of [30] with a change of the axes so that the local behaviors of S 2 and S 3 visually correspond to the local behaviors of the trajectories in Fig 16 (b) around the The connecting manifold L is also called the 'slow manifold' in [30] due to the fact that trajectories move slower when passing near it. As trajectories reach S 3 , the eigenvector corresponding to the real eigenvalue of S 3 breaks out of the xz-plane and redirects the flow toward S 2 . As shown in Fig 16 (a) and (b), as B/A = 1 moves to B/A > 1, this process transforms each spherical shell to a toroidal shell. The solutions scroll down the toroidal surfaces until a limit cycle (shown in green in Fig 16 (b)) is reached. It is worth pointing out that this limit cycle is a torus of 0-diameter and corresponds to the sphere of 0-diameter, namely, the central steady point of L also shown in green in Fig 16 (a). However, as the authors elaborate in [31], while for B/A = 1 the entire positive space is filled with nested spheres, when B/A > 1, only spheres up to a certain volume become tori. More specifically, quoting the authors: "to preserve uniqueness of solutions, the connections through the slow manifold L are made in a way that higher volume shells require slower, or higher resolution, trajectories within the bundle". As they further explain, to connect all shells through L, (Σ) would need to possess an infinite resolution.
As this is never the case, the solutions evolving on shells of higher volume are 'choked' by the slow manifold. This generates solution indetermination, which forces higher volume shells to rapidly collapse or dissipate. The behavior stabilizes when trajectories enter the region where the choking becomes weak and weak chaos appears. As shown in both [30] and [31], the outermost shell of the toroidal nesting is a fractal torus. Note that in  nesting. By zooming on the slow manifold of the outermost fractal torus shell, in Fig. 17 (b) we can view the 'hole drilling' behavior of the trajectories.
As already mentioned, as B/A = 1 changes to B/A > 1, S 2 changes from an unstable center to an inward unstable vortex and S 3 changes from a stable center to an outward stable vortex. It is worth reminding that this change in local behavior is true not only for the specific parametrical region simulated in Fig. 16 and 17 , but applies to all cases satisfying (1/8B − 1) A/B < C ≤ 2(1 + √ 2). For details we refer the reader to Tables II and III in [30] that recapitulate the extensive diagrammatic analysis done therein.
Finally, it is worth observing the changing of the local behavior around S 2 and S 3 in our numerical simulations. In Fig 16 (a) For example, as mentioned in [31], higher resolution produces a larger fractal torus and a finer connecting manifold. However, the 'hole drilling' process and the creation of a toroidal nesting is always a common feature.

Connecting the dynamical system with solid 2-dimensional 0-surgery
In this section, we will focus on the process of solid 2-dimensional 0-surgery on a 3-ball D 3 viewed as a continuum of concentric spheres together with their common center: Recall from Section 6.2 that the process is defined as the union of 2-dimensional 0-surgeries on the whole continuum of concentric spheres S 2 r and on the limit point P . For each spherical layer, the process starts with attracting forces acting between S 0 × D 2 , i.e two points, or poles, centers of two discs.
Having presented the dynamical system (Σ) in Section 7.1 and its local behavior in Section 7.2, its connection with solid 2-dimensional 0-surgery on a 3-ball is now straightforward. To be precise, surgery is performed on the manifold formed by the trajectories of (Σ). Indeed, as seen in Fig 16 (a)  shown in green in in Fig 16 (b) is a limit cycle. In other words, surgery on the limit point P creates the limit cycle. As mentioned in [31], this type of bifurcation is a 'Hopf bifurcation', so surgery on the trajectories can be also seen as a Hopf bifurcation.
Hence, instead of viewing surgery as an abstract topological process, we may now view it as a property of a dynamical system. Moreover, natural phenomena exhibiting 2-dimensional topological surgery through a 'hole-drilling' process, such as the creation of Falaco solitons, the formation of tornadoes, of whirls, of wormholes, etc, may be modeled mathematically by the dynamical system (Σ). This system enhances the topological model presented in Fig. 14  where the notion of embedded surgery in 3-space is introduced. By 3-space we mean here the compactification of R 3 which is the 3-sphere S 3 . This choice, as opposed to R 3 , takes advantage of the duality of the descriptions of S 3 . In this section we present the three most common descriptions of S 3 (see Section 8.1) in which this duality is apparent and which will set the ground for defining the notion of embedded surgery in S 3 (see Section 9). Beyond that, in Section 8.2, we also demonstrate how these descriptions are interrelated. Finally, in Section 8.3, we pin down how the trajectories of the dynamical system (Σ) presented in Section 7 are related to the descriptions of S 3 and further introduce a Hamiltonian system exhibiting the topology of S 3 .

Descriptions of S 3
In dimension 3, the simplest c.c.o. 3-manifolds are: the 3-sphere S 3 and the lens spaces L(p, q). In this analysis however, we will focus on S 3 . We start by recalling its three most common descriptions:  Note that, in both cases B 3 represents the hole space outside D 3 which means that the spherical nesting of B 3 in Fig. 18 (b') extends to infinity, even though only a subset of B 3 is shown. This is another way of viewing R 3 as the de-compactification of S 3 . This picture is the analogue of the stereographic projection of S 2 on the plane R 2 (recall Figure 2), whereby the projections of the concentric circles of the south hemisphere together with the projections of the concentric circles of the north hemisphere form the well-known polar description of R 2 with the unbounded continuum of concentric circles.

Via two solid tori
The third well-known representation of S 3 is as the union of two solid tori along their common boundary: S 3 = V 1 ∪ ϑ V 2 , via the torus homeomorphism ϑ along the common boundary, see Fig. 18 (c). ϑ maps a meridian of V 2 to a longitude of V 1 which has linking number zero with the core curve c of V 1 . The illustration in Fig. 18 (c) gives an idea of this splitting of S 3 . In the figure, the core curve c of V 1 is in dashed black. So, the complement of a solid torus V 1 in S 3 is another solid torus V 2 whose core curve l (shown in dashed red) may be assumed to pass by the point at infinity. Note that, S 3 minus the core curves c and l of V 1 and V 2 can be viewed as a continuum of nested tori, see Fig. 18 (c').
When removing the point at infinity in the representation of S 3 as a union of two solid tori, the core of the solid torus V 2 becomes an infinite line l and the nested tori of V 2 can now be seen wrapping around the nested tori of V 1 , see the passage from Fig. 18 (c) to Fig. 18 (c'). Therefore, R 3 can be viewed as an unbounded continuum of nested tori, together with the core curve c of V 1 and the infinite line l. This line l joins pairs of antipodal points of all concentric spheres of the first description. Note that in the nested spheres description (Fig. 18 (b')) the line l pierces all spheres while in the nested tori description the line l is the 'untouched' limit circle of all tori.

Remark 7.
It is also worth mentioning that another way to visualize S 3 as two solid tori is the Hopf fibration, which is a map of S 3 into S 2 . The parallels of S 2 correspond to the nested tori of S 3 , the north pole of S 2 correspond to the core curve l of V 2 while the south pole of S 2 corresponds to the core curve c of V 1 . An insightful animation of the Hopf fibration can be found in [10].

Via corking
The connection between the first two descriptions of S 3 was already discussed in previous section. The third description is a bit harder to connect with the first two. We shall do this here. A way to see this connection is the following. Consider the description of S 3 as the union of two 3-balls, B 3 and D 3 (Fig. 18 (b')). Combining with the third description of S 3 (Fig. 18 (c')) we notice that both 3-balls are pierced by the core curve l of the solid torus V 2 . Therefore, D 3 can be viewed as the solid torus V 1 to which a solid cylinder D 1 × D 2 is attached via the homeomorphism ϑ: This solid cylinder is part of the solid torus V 2 , a 'cork' filling the hole of V 1 . Its core curve is an arc L, part of the core curve l of V 2 . View Fig 19 (a). The second ball B 3 ( Fig. 19 (b)) can be viewed as the remaining of V 2 after removing the 'cork' D 1 × D 2 : In other words the solid torus V 2 is cut into two solid cylinders, one comprising the 'cork' of V 1 and the other comprising the 3-ball B 3 . Figure 19. Passing from (a) S 3 as two solid tori to (b) S 3 as two balls.
Remark 8. If we remove a whole neighborhood B 3 of the point at infinity and focus on the remaining 3-ball D 3 , the line l of the previous picture is truncated to the arc L and the solid cylinder V 2 is truncated to the 'cork' of D 3 .
Remark 9. This arc L corresponds to the segment L joining the steady state points of the dynamical system of Section 7.

Via surgery
We can also pass from the two-ball description to the two-tori description of

Dynamical systems exhibiting the topology of S 3
In this section, we go back to the 3-dimensional Lotka-Volterra system (Σ) and see how its trajectories relate to the descriptions of S 3 and further present a 4-dimensional Hamiltonian system exhibiting the topology of S 3 .

The 3-dimensional Lotka-Volterra system
We will now pin down how the trajectories of (Σ) presented in Section 7 relate to the topology of S 3 . We start with the spherical nesting of Fig 16 (a) which can be viewed as the 3-ball D 3 shown in Fig. 18 (b) and (b'). Surgery on its central point creates the limit cycle which is the core curve c of V 1 shown in Fig. 18 (c) and (c'). If we extend the spherical shells of Fig 16 to all of R 3 and assume that the entire nest resolves to a toroidal nest, then the slow manifold L becomes the infinite line l. In the two-ball description of S 3 , l pierces all spheres, recall Fig. 18 (b'), while in the two-tori description, it is the core curve of V 2 or the 'untouched' limit circle of all tori, recall Fig. 18 (c) and (c'). Given that H is the sum of the energy functions of two harmonic oscillators with frequencies m and n and assuming that H is at least C 1 , m > 0 and n > 0, the time evolution of this system is given by the following system of four ODE of Hamilton:
We will now identify the topology of S 3 in Figure 20. The two critical circle S 1 h + and S 1 h − mentioned above are the core curve c of V 1 and the infinite line l respectively, see Fig. 18 (c) and (c'). While l can be seen in both Figure 18 and Figure 20, the core curve c can only be seen if we visualize the trajectories of Fig. 20 step by step. Indeed, in Fig. 21 (a) we see the core curve c which is quickly covered by the other trajectories. The first three toroidal layers are shown in Fig. 21 (b),(c) and (d). Note that, comparing the trajectories of Fig. 21 with those of (Σ) shown in Fig 16 (b), a key difference is that here, as opposed to system (Σ), the nesting of tori does extend to infinity. To close the topological analogy, one can visualize S 3 = V 1 ∪ ϑ V 2 by considering that V 1 is any finite number of toroidal nestings. For example V 1 could be any of the tori shown in Fig. 21 (b),(c) or (d) while, in each of these case, V 2 is naturally defined as the complement space.

Embedded surgery
In this section we will examine how the ambient space can be involved in the process of surgery and introduce the notion of embedded surgery in order to model such phenomena.
As we will see, depending on the dimension of the manifold, the ambient space either leaves 'room' for the initial manifold to assume a more complicated configuration or it participates more actively in the process. Independently of dimensions, embedding surgery has the advantage that it allows us to view surgery as a process happening inside a space instead of abstractly. We define it as follows: Since in this analysis we focus on phenomena exhibiting embedded 1-and 2-dimensional surgery in 3-space, from now on we fix d = 3 and, for our purposes, we consider S 3 or R 3 as our standard 3-space.

Embedded 1-dimensional surgery
In dimension 1, the notion of embedded surgery allows the topological modeling of phenomena with more complicated initial 1-manifolds. Let us demonstrate this with the example of site-specific DNA recombination. In this process, the initial manifold  The first electron microscope picture of knotted DNA was presented in [34]. In this experimental study, we see how genetically engineered circular DNA molecules can form DNA knots and links through the action of a certain recombination enzyme. A similar picture is presented in Fig. 22, where site-specific recombination of a DNA molecule produces the Hopf link. It is worth mentioning that there are infinitely many knot types and that 1-dimensional 0-surgery on a knot may change the knot type or even result in a two-component link (as shown in Fig. 22). Since a knot is by definition an embedding of M = S 1 in S 3 or R 3 , in this case embedded 1-dimensional surgery is the so-called knot surgery. A good introductory book on knot theory is [1] among many others.
We can summarize the above by stating that for M = S 1 , embedding in S 3 allows the initial manifold to become any type of knot. More generally, in dimension 1 the ambient space which is of codimension 2 gives enough 'room' for the initial 1-manifold to assume a more complicated homeomorphic configuration.
Remark 10. Of course we also have, in theory, the notion of embedded solid 1-dimensional 0-surgery whereby the initial manifold is an embedding of a disc in 3-space.

Embedded 2-dimensional surgery
Passing now to 2-dimensional surgeries, let us first note that an embedding of a sphere M = S 2 in S 3 presents no knotting because knotting requires embeddings of codimension 2. However, in this case the ambient space plays a different role. Namely, embedding 2-dimensional surgeries allows the complementary space of the initial manifold to participate actively in the process. Indeed, while some natural phenomena undergoing surgery can be viewed as 'local', in the sense that they can be considered independently from the surrounding space, some others are intrinsically related to the surrounding space. This relation can be both causal, in the sense that the ambient space is involved in the triggering of the forces causing surgery, and consequential, in the sense that the forces causing surgery can have an impact on the ambient space in which they take place.
As mentioned in the introduction of Section 6, in most natural phenomena that exhibit 2-dimensional surgery, the initial manifold is a solid 3-dimensional object. Hence, in the next sections, we describe natural phenomena undergoing solid 2-dimensional surgeries which exhibit the causal or consequential relation to the ambient space mentioned above and are therefore better described by considering them as embedded in S 3 or in R 3 . In parallel, we describe how these processes are altering the whole space S 3 or R 3 .

Modeling phenomena exhibiting embedded solid 2-dimensional surgery
In each of the following sections a natural phenomena undergoing embedding solid 2dimensional surgery is analyzed. As we will see, the topological considerations of these processes also have physical implications.

A topological model for the density distribution in black hole formation
Let us start by considering the density distribution in black hole formation. Most black holes are formed from the remnants of a large star that dies in a supernova explosion.
Their gravitational field is so strong that not even light can escape. In the simulation of a black hole formation in [25], the density distribution at the core of a collapsing massive star is shown. Fig. 23 (2) shows three instants of this simulation, which indicate that matter performs solid 2-dimensional 0-surgery as it collapses into a black hole. In fact, matter collapses at the center of attraction of the initial manifold M = D 3 creating the singularity, that is, the center of the black hole (shown as a black dot in instance (c) of Fig. 23 (2)), which is surrounded by the toroidal accretion disc (shown in white in instance (c) of Fig. 23 (2)). Let us be reminded here that an accretion disc is a rotating disc of matter formed by accretion.
Note now that the strong gravitational forces have altered the space surrounding the initial star and that the singularity is created outside the final solid torus. This means  that the process of surgery in this phenomenon has moreover altered matter outside the manifold in which it occurs. In other words, the effect of the forces causing surgery propagates to the complement space, thus causing a more global change in 3-space. This fact makes black hole formation a phenomenon that topologically undergoes embedded solid 2-dimensional 0-surgery.
From the descriptions of S 3 in Section 9.2, it becomes apparent that embedded solid 2-dimensional 0-surgery on one 3-ball describes the passage from the two-ball description to the two-solid tori description of S 3 . This can be seen in R 3 in instances (a) to (c) of Fig. 23 (1) but is more obvious by looking at instances (a) to (c) of Fig. 24 which show the corresponding view in S 3 .
We will now detail the instances of the process of embedded solid 2-dimensional 0surgery on M = D 3 by referring to both the view in S 3 and the corresponding decompacified view in R 3 . Let M = D 3 be the solid ball having arc L as a diameter and the complement space be the other solid ball B 3 containing the point at infinity; see instances (a) of Fig. 24 and (a) of Fig. 23. Note that, in both cases B 3 represents the hole space outside D 3 which means that the spherical nesting of B 3 in instance Fig. 23 (a) extends to infinity, even though only a subset of B 3 is shown. This joining arc L is seen as part of a simple closed curve l passing by the point at infinity. In instances (b) of Fig. 24 and (b) of Fig. 23, we see the 'drilling' along L as a result of the attracting forces. This is exactly the same process as in Fig. 14 (b 1 ) if we restrict it to D 3 . But since we have embedded the process in S 3 or R 3 , the complement space B 3 participates in the process and, in fact, it is also undergoing solid 2-dimensional 0-surgery. Indeed, the 'matter' that is being drilled out from the interior of D 3 can be viewed as 'matter' of the outer sphere B 3 invading D 3 . In instances (c) of Fig. 24 and (c) of Fig. 23, we can see that, as surgery transforms the solid ball D 3 into the solid torus V 1 , B 3 is transformed into V 2 . That is, the nesting of concentric spheres of D 3 (respectively of B 3 ) is transformed into the nesting of concentric tori in the interior of V 1 (respectively of V 2 ). The point at the origin (in green), which is also the attracting center, turns into the core curve c of V 1 (in green) which, by Definition 4 is 2-dimensional 0-surgery on a point. As seen in instance (c) of Fig. 24 and (c) of Fig. 23 (1), the result of surgery is the two solid tori V 1 and V 2 forming S 3 .
The described process can be viewed as a double surgery resulting from a single attracting center which is inside the first 3-ball D 3 and outside the second 3-ball B 3 . This attracting center is illustrated (in blue) in instance (a) of Fig. 23 but also in (a) of Fig. 24, where it is shown that the colinear attracting forces causing the double surgery can be viewed as acting on D 3 (the two blue arrows) and also as acting on the complement space B 3 (the two dotted blue arrows), since they are applied on the common boundary of the two 3-balls. Note that in both cases, the attracting center coincides with the limit point of the spherical layers that D 3 is made of, that is, their common center and the center of This process is the embedded analog of the solid 2-dimensional 1-surgery on a solid torus D 2 × S 1 defined in Definition 5 and shown in Fig. 14 (b 1 ) in reverse order. Here too, the process can be viewed as a double surgery resulting from one attracting center which is outside the first solid torus V 1 and inside the second solid torus V 2 . This attracting center is illustrated (in orange) in instance (c) of Fig. 24 where it is shown that the coplanar forces causing surgery are applied on the common boundary of V 1 and V 2 and can be viewed as attracting forces along a longitude when acting on V 1 and as attracting forces along a meridian when acting on the complement space V 2 .
Indeed, if one looks at the density distribution during the formation of a black hole and examines it as an isolated event in space, this process shows a decompactified view of the passage from a two 3-ball description of S 3 , that is, the core of the star and the surrounding space, to a two solid tori description, namely the toroidal accretion disc surrounding the black hole (shown in white in instance (c) of Fig. 23 (2)) and the surrounding space.
Finally, it is worth pinning down the following spatial duality of embedded solid 2-dimensional 0-surgery for M = D 3 : the attraction of two points lying on the boundary of segment L by the center of D 3 can be equivalently viewed in the complement space as the repulsion of these points by the center of B 3 (that is, the point at infinity) on the boundary of the segment l − L (or the segments, if viewed in R 3 ). Hence, the aforementioned duality tells us that the attracting forces from the attracting center that are collapsing the core of the star can be equivalently viewed as repelling forces from the point at infinity lying in the surrounding space.

A topological model for the formation of tornadoes
Another example of global phenomenon is the formation of tornadoes, recall Fig. 10 (2). As mentioned in Section 6 this phenomenon can be modelled by solid 2-dimensional 0-surgery. However, here, the initial manifold is different than D 3 . Indeed, if we consider a 3-ball around a point of the cloud and another 3-ball around a point on the ground, then the initial manifold is M = D 3 × S 0 and the process followed is the one shown in Fig. 14 (b 2 ) (from right to left). More precisely, if certain meteorological conditions are met, an attracting force between the cloud and the earth beneath is created. This force is shown in blue in see Fig. 25 (1). Then, funnel-shaped clouds start descending toward the ground, see Fig. 25 (2). Once they reach it, they become tornadoes, see Fig. 25 (3). The only difference compared to our model is that here the attracting center is on the ground, see Fig. 25 (1), and only one of the two 3-balls (the 3-ball of cloud) is deformed by the attraction. This lack of symmetry in the process can be obviously explained by the big difference in the density of the materials.
During this process, a solid cylinder D 2 × S 1 containing many layers of air is created. Each layer of air revolves in a helicoidal motion which is modeled using a twisting embedding as shown in Fig. 10 (1) (for an example of a twisting embedding, the reader is referred Section 3.2). Although all these layers undergo local dynamic 2-dimensional 0-surgeries which are triggered by local forces (shown in blue in Fig. 25 (1)), these local forces are not enough to explain the dynamics of the phenomenon. Indeed, the process is triggered by the difference in the conditions of the lower and upper atmosphere which create an air cycle. This air cycle lies in the complement space of the initial manifold M = D 3 × S 0 and of the solid cylinder D 2 × S 1 , but is also involved in the creation of the funnel-shaped clouds that will join the two initial 3-balls. Therefore in this phenomenon, surgery is the outcome of global changes and this fact makes tornado formation an example of embedded It is worth mentioning that the complement space containing the aforementioned air cycle is also undergoing solid 2-dimensional 0-surgery. The process can be seen in R 3 in instances (a) to (d) of Fig. 26 while the corresponding view in S 3 is shown in instances

Decompactified 2-dimensional surgery
We present here the notion of decompactified 2-dimensional surgery which allows us to visualize 2-dimensional surgery in R 2 instead of R 3 .
Let us first recall from Section 4 that an m-dimensional n-surgery happens inside the handle D n+1 × D m−n . In Fig. 27  To grasp this last observation, recall that S 1 is obtained by a 180°rotation of S 0 and S 2 is obtained by a 180°rotation of S 1 . Moving up one dimension, since S 2 is embedded in R 3 , the S 3 created by rotation requires a fourth dimension in order to be visualized.
In order to overcome this difficulty we project stereographically S 2 in R 2 , as shown in Note that, in analogy to 2-dimensional 0-surgery, decompactified 2-dimensional 0surgery can also be seen as a process caused by attracting forces and an attracting center.
The forces are not shown here in order to keep the figures lighter.
In Section 10.2 we will rotate the instances of decompactified 2-dimensional 0-surgery in order to obtain our first visualization of 3-dimensional surgery in R 3 .
Remark 13. The decompactified 2-dimensional 1-surgery can also be defined in analogy to the decompactified 2-dimensional 0-surgery but it is simpler to view it as its reverse process.

Visualizing 3-dimensional surgery in R 3
As mentioned in Section 10.1, rotating the S 2 made of the initial and final instances of 2-dimensional surgery gives us the S 3 = ∂D 4 made of the initial and final instances of 3-dimensional surgery. We will now rotate the stereographic projection of S 2 in R 2 , see Fig. 28, to obtain the stereographic projection of the initial and final instances of 3-dimensional surgery in R 3 . We will discuss the two processes of 3-dimensional surgery, namely 3-dimensional 1-surgery and 3-dimensional 0-surgery, which were introduced in

Initial and final steps
Let us recall the initial and final instances of our two processes of 3-dimensional surgery.
For 3-dimensional 1-surgery, the initial and final instances are solid tori S 1 × D 2 and D 2 × S 1 while for 3-dimensional 0-surgery, we have two 3-balls S 0 × D 3 and a thickened sphere D 1 × S 2 .
Rotating our decompactified view in R 2 by 180°vertically gives us the initial and final instances of 3-dimensional 1-surgery in R 3 . In this case the axis of rotation is line l which is at equal distance from the two flattened discs and is shown in green in Fig. 29 (a). We can directly see that this rotation transforms the two discs S 0 × D 2 (the first instance of decompactified 2-dimensional 0-surgery) to the solid torus S 1 1 × D 2 (the first instance of 3-dimensional 1-surgery), see Fig. 29 (b). Each of the arcs connecting the two discs S 0 × D 2 generates through the rotation a 2-dimensional disc, the set of all such discs being parametrized by the points of the line l in R 3 . Therefore the complement of the solid torus S 1 1 × D 2 is another solid torus D 2 × S 1 2 , see Fig. 29 (b), where line l in R 3 is circle the core curve c of V 1 is S 1 1 and the core curve Similarly, rotating our decompactified view in R 2 by 180°horizontally gives us the initial and final instances of 3-dimensional 0-surgery in R 3 . The axis of rotation is line l which pierces the two flattened discs and is shown in grey in Fig. 29 (a). We can directly see that this rotation transforms the two discs S 0 × D 2 (the first instance of decompactified 2-dimensional 0-surgery) to two 3-balls S 0 1 × D 3 (the first instance of 3-dimensional 0-surgery), see Fig. 29 (c). The rotation of line l along l creates a plane that cuts through R 3 and separates the two resulting 3-balls S 0 1 × D 3 . This plane is thickened by the arcs connecting the two discs S 0 × D 2 which have also rotated, see Fig. 29 (c). This plane is the decompactified view of sphere S 2 2 in S 3 which can be viewed as a rotation of circle l ∪ {∞} = S 1 2 . Therefore the complement of the two 3-balls S 0 1 × D 3 is a thickened sphere D 1 × S 2 2 where the plane resulting from the rotation of line l in R 3 is sphere S 2 2 in In both cases, in Fig. 29 (a), S 3 is represented as the result of rotating the 2 sphere

Intermediate steps
We are now ready to visualize the intermediate steps of both types of 3-dimensional surgery. By rotating the instances of decompactified 2-dimensional 0-surgery (shown in In these two processes, as in lower dimensional surgeries, the time-evolution of surgery passes through a singular point. Namely, for 3-dimensional 1-surgery we see a solid torus S 1 1 × D 2 collapsing to a singularity from which emerges the complement solid torus D 2 × S 1 2 . This, if visualized in R 3 , fills the rest of the space, see Fig. 30 (a). In this case we have used the standard (identity) embedding of S 1 1 × D 2 denoted by h s , which induces a 'gluing' homeomorphism along the common boundary S 1 × S 1 , such that the meridians of solid torus V 1 = S 1 1 × D 2 are mapped to the longitudes of solid torus V 2 = D 2 × S 1 2 . In other For 3-dimensional 0-surgery, we see the two 3-balls S 0 1 × D 3 collapsing to a singularity from which emerges the thickened sphere D 1 × S 2 2 which, if decompactified in R 3 , is a thickened plane filling the rest of the space, see Fig. 30 (b).

The continuity of 3-dimensional surgery
In this section we analyze the concept of continuity for 3-dimensional surgery.
Let us first recall from Section 4 that all types of 3-dimensional surgery take place inside the 4-dimensional handle D n+1 × D 3−n , n < 3 and that the processes of both 3-dimensional 0-and 1-surgery can be viewed as taking the boundary of the first factor D n+1 , thickening it, passing through the unique intersection point D n+1 ∩ D 3−n and then letting the thickened boundary of the second factor D 3−n emerge. We will first present the core view which shows how we pass from the boundary of D n+1 to the boundary of D 3−n . We will then apply the different kinds of thickenings (or framings) to the cores in order to illustrate both processes in R 4 .
More precisely, 3-dimensional 1-surgery takes place inside D 2 1 × D 2 2 . In this case, we go from the core S 1 1 = ∂D 2 1 to the core S 1 2 = ∂D 2 2 by passing through the unique intersection D 2 1 ∩ D 2 2 . As mentioned in Section 2.4, we consider that n-balls are centered at the origin.
We will now thicken the aforementioned cores in order to present our illustrations in were filling the rest of the space in R 3 . Our goal here is to obtain the corresponding undistorted illustrations in R 4 .
For 3-dimensional 1-surgery, we start by thickening our core view S 1 1 shown in Fig. 31 (a 1 ) with a D 2 . We collapse S 1 1 × D 2 to a singularity and then need to uncollapse it in a way that produces the complement solid torus D 2 × S 1 2 as a thickening of its core S 1 2 . The whole process is shown in Fig. 31 (a 2 ).
For 3-dimensional 0-surgery, we start by thickening our core view S 0 1 shown in Fig. 31 (b 1 ) with a D 3 . We collapse S 0 1 × D 3 to a singularity and then we need to uncollapse it in a way that produces the D 1 × S 2 2 as a thickening of its core sphere S 2 2 .
The whole process is shown in Fig. 31 (b 2 ).
In both cases, we have combined the core with the R 4 view and added the corresponding  Fig. 27  . Indeed, starting with cylinder D 1 × S 1 , we first decompactify S 1 to R 1 to obtain D 1 × R 1 , see Fig. 32 (a). Then, using the same rotational axis as the one described in Section 10.2.1 and shown in green in Fig. 32 (a), we obtain the decompactified D 2 × R 1 , where each segment D 1 has been rotated to a disc D 2 . We finally recompactify R 1 to S 1 to obtain D 2 × S 1 . Note that the axis starts as a circle, becomes straight during decompactification and becomes a circle again when we recompactify. Similarly, starting again with the D 1 × S 1 of 2-dimensional 0-surgery, we first decompactify S 1 to R 1 to obtain D 1 × R 1 , see Fig. 32 (b). Then, using the same rotational axis as the one described in Section 10.2.1 and shown in grey Fig. 32 (b), we obtain the decompactified D 1 × R 2 where each line R 1 has been rotated to a plane R 2 . In Fig. 32 (b), we have used an oblique view of both D 1 × R 1 and D 1 × R 2 so the effect of the rotation can be visible. We finally compactify R 2 to S 2 to obtain the thickened sphere (or hollow This process of decompactifying, rotating and compactifying again allowed us to visualize the final instances of 3-dimensional 1-and 0-surgery in relation with our initial visualization of D 1 × S 1 of 2-dimensional 0-surgery by following a reasoning similar to the one used in Section 10.2 for the visualizations in R 3 . The difference being that in the visualizations of Section 10.2, D 1 × S 1 was obtained as part of the decompactification of S 2 , hence it was inevitably deformed so that its union with S 0 × D 2 would form R 2 .
As this constrain does not exist here, when S 1 is decompactified to R 1 , its framing D 1 follows without undergoing such deformation, see the passage from the first to the second instance in Fig. 32 (a) and (b).

Modeling black hole formation from cosmic strings
In this section we will see how the formation of black holes from cosmic strings can be modeled by 3-dimensional 1-surgery.

Terminology
We will first explain the terms of Schwarzschild radius, event horizon and gravitational singularity which will be used in the following sections.
The Schwarzschild radius is the radius of a 2-sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface of the sphere would equal the speed of light. If anything collapses to or below this radius, a black hole is formed. The event horizon is a boundary in spacetime beyond which events cannot affect an outside observer and is most commonly associated with black holes. For a nonrotating black hole, the Schwarzschild radius delimits a spherical event horizon.
In the center of a black hole, general relativity predicts the existence of a gravitational singularity (or space-time singularity), i.e a region in space in which matter takes infinite density and 0 volume (basically infinitely dense and infinitely small). The singularity cannot be seen as it is covered by the event horizon.

Black holes from cosmic strings
Cosmic strings are hypothetical 1-dimensional topological defects which may have formed in the early universe and are predicted by both quantum field theory and string theory models. Their existence was first contemplated by Tom Kibble in the 1970s. In [9], S.W.
Hawking estimates that a fraction of cosmic string loops can collapse to a small size inside their Schwarzschild radius thus forming a black hole. As he mentions, under certain conditions, 'one would expect an event horizon to form, and the loop to disappear into a black hole'.
Note that other estimations of the fraction of cosmic string loops which collapse to form black holes have been made in subsequent work, see [4] and [22]. Topologically, the aforementioned loop can be considered to be a solid torus S 1 × D 2 embedded in an initial manifold M . The thickening D 2 can be considered to be very small, as the diameter of a cosmic strings is of the same order of magnitude as that of a proton, i.e.
≈ 1 fm or smaller. Further, we consider M as being the 3-space S 3 or R 3 or a 3-manifold corresponding to the 3-dimensional spatial section of the 4-dimensional space-time of the universe. The loop S 1 × D 2 collapses to a small size inside its Schwarzschild radius thus creating a black hole the center of which contains the singularity. In this scenario, the inital 3-space M becomes a singular manifold at that point.
Physicists are undecided whether the prediction of this singularity means that it actually exists or that current knowledge is insufficient to describe what happens at such extreme density. As we will see in the next section, we can avoid this singularity by considering that the collapsing of a cosmic string loop is followed by the uncollapsing of another cosmic string loop. In other words, we propose that the creation of a black hole is a 3-dimensional 1-surgery which changes the initial 3-dimensional space M to another 3-dimensional space χ(M ) by passing through a singular point. As detailed in Section 10.3, the time evolution of this process happens locally inside the handle D 2 × D 2 and requires four spatial dimensions in order to be visualized but each 'slice' of the process is a 3-dimensional manifold.
Note that this process is different from the global process of embedded surgery used to model the density distribution of black hole formation in Section 9.3 as it describes the local process of a cosmic string collapsing to a black hole inside the event horizon.

Black holes from 3-dimensional 1-surgery
We will now describe the process of 3-dimensional 1-surgery on M step by step. We start with an embedding of the loop S 1 × D 2 , which may also be knotted. Using an analogue which is two dimensions lower, M is shown as a line while the core S 1 of the embedding S 1 × D 2 is shown as the core S 0 of embedding S 0 × D 1 . In Fig. 33 (initial), core S 0 is shown in red and its thickening in grey. Since the process of surgery is a local process, black hole formation can be seen independently of M . Therefore we zoom in to see this local procedure in instances (1) to (4) of Fig. 33 which happens inside D 2 × D 2 .
As mentioned in Section 5 and throughout this analysis, the local process of surgery is considered as a result of attracting forces. In the 3-dimensional case, we deliberately didn't show these forces in Fig. 31 in order to keep the illustrations lighter but as explained in Section 5.5, we know that the forces of our model are applied to the core 1-embedding e = h | : S 1 = S 1 × {0} → M of the framed n-embedding h : S 1 × D 2 → M .
These forces are added in blue in Fig. 33 (1), where we see the same process as Fig. 31 (a 2 ) but with a knotted embedding of the core S 1 . Note that these local forces of our model correspond to the string tension, which collapses the cosmic string (see [9] for details). In instance (2) of Fig. 33 the cosmic string shrinks to a radius smaller than its Schwarzschild radius, thus the event horizon is formed. We are not showing the black hole inside the event horizon as we want to focus on the topological change. Further, instance  According to our model, after the collapsing the process doesn't stop, but another manifold D 2 ×S 1 , which corresponds to another cosmic string loop, grows from the singular point of instance Fig. 33 (3), and this is the added value of our model. In Fig. 33 (4) we show the uncollapsing of cosmic string D 2 × S 1 which transforms the initial manifold M to χ(M ) = M \ h(S 1 × D 2 ) ∪ h (D 2 × S 1 ), see Fig. 33 (final). Note that instances It is worth mentioning that Fig. 33 (final) is also shown two dimensions lower. Namely, as in Fig. 33 (initial), the core S 1 of the cosmic string loop D 2 × S 1 is represented by the core S 0 of D 1 × S 0 . In Fig. 33 (final), core S 0 is shown in green and its thickening in grey. The whole process occurs inside the handle D 1 × D 1 illustrated in the upper part of Fig. 33 which, following this analogy, stands for D 2 × D 2 . The global 4-dimensional visualization of 3-dimensional 1-surgery on an initial 3-manifold such as S 3 following the line of thought of Sections 10.2.2 and 10.3 is an intriguing subject and will be the subject of future work. However, for the purpose of this analysis we do not need to visualize the initial and final manifold but rather the idea behind the local process illustrated in instances Fig. 33 (1),(2),(3),(4).
Summarizing the above, modeling the collapsing of a cosmic string loop with a 3dimensional 1-surgery allows us to go through the singular point of the black hole without having a a singular manifold in the end. Instead, we end up in the same universe with a local topology change from the 3-dimensional space M to the 3-dimensional space χ(M ) and, as seen in Fig. 33 (2),(3),(4), this topology change happens within the event horizon.

Conclusions
In this thesis we explained many natural processes via topological surgery. Examples Finally, we also modeled the formation of black holes from cosmic strings using 3dimensional 1-surgery. As our model suggests that a black hole does not necessarily result in a spatial singularity, it would be very interesting to collaborate with physicists in order to investigate the physical implications of the proposed topological change.
We hope that through this study, topology and dynamics of natural phenomena, as well as topological surgery itself, will be better understood and that our connections will serve as ground for many more insightful observations and new physical implications.
Manifolds 1. A Hausdorff space M n with countable base is said to be an n-dimensional topological manifold if any point x ∈ M n has a neighborhood homeomorphic to R n or to R n + , where R n + = {(x 1 , ..., x n ) | x i ∈ R, x 1 ≥ 0}. For example, a surface is a 2-dimensional manifold.
2. The set of all points x ∈ M n that have no neigbourhoods homeomorphic to R n is called the boundary of the manifold M n and is denoted by ∂M n . When ∂M n = ∅, we say that M n is a manifold without boundary. It is easy to verify that if the boundary of a manifold M n is nonempty, then it is an (n − 1)-dimensional manifold.

Topologies
3. If (X, τ ) is a topological space, a base of the space X is a subfamily τ ⊂ τ such that any element of τ can be represented as the union of elements of τ . In other words, τ is a family of open sets such that any open set of X can be represented as the union of sets from this family. In the case when at least one base of X is countable, we say that X is a space with countable base.

4.
To define the topology τ , it suffices to indicate a base of the space. For example, in the space R n = {(x 1 , ..., x n ) | x i ∈ R}, the standard topology is given by the base U a, = {x ∈ R n | |x − a|< }, where a ∈ R n and > 0. We can additionally require that all the coordinates of the point a, as well as the number , be rational; in this case we obtain a countable base.

5.
To the set R n let us add the element ∞ and introduce in R n ∪ {∞} the topology whose base is the base of R n to which we have added the family of sets U ∞,R = {x ∈ R n | |x|> R} ∪ {∞}. The topological space thus obtained is called the one-point compactification of R n ; it can be shown that this space is homeomorphic to the n−dimensional sphere S n = {x ∈ R n+1 | |x|= 1}.