On knotoids, braidoids and applications

Abstract

In this present thesis, we study on the theory of knotoids that was introduced by V. Turaev in 2012, we introduce the theory of braidoids and lastly we apply the theory of knotoids to the study of proteins. In the first chapter of the thesis, after a detailed recollection of basic notions of knotoids, we construct new invariants of knotoids, including the arrow polynomial, the odd writhe, the parity bracket polynomial, the affine index polynomial, and also give an introduction to the theory of virtual knotoids. These invariants are defined for both classical (knotoids in $S^2$ or $\mathbb{R}^2$) and virtual knotoids in analogy to the corresponding invariants of virtual knots. We discuss on the virtual closure map that connects the classical knotoid theory to the virtual knot theory and show it is a non-injective and non-surjective map. We show that the arrow polynomial that is an oriented generalization of the bracket polynomial, provides a lower bound estimation for the height (or complexity) invariant of knotoids. We then introduce the affine index polynomial of knotoids and show that the affine index polynomial of a knotoid in $S^2$ is symmetric. We provide one more lower bound estimation for the height invariant via the affine index polynomial. We compare the two lower bounds for the height invariant provided by the arrow polynomial and the affine index polynomial with some examples. Additionally, we observe that knotoids are the first knotted objects given in the classical setting that admit a non-trivial parity. We introduce parity invariants using the parity defined; such as the odd writhe and the parity bracket polynomial. We also give a geometric interpretation of planar knotoids in terms of open ended space curves. This interpretation later in the last chapter is used for the study of protein chains. This part covers the results of works with Kauffman. In the second chapter, we introduce the theory of braidoids that forms a `braided' counterpart theory for the theory of knotoids. We introduce notions of braidoid diagram and isotopy classes of braidoid diagrams, namely, braidoids, and we define a closure operation on a special class of braidoids namely labeled braidoid diagrams. We give two algorithms to turn a knotoid or a multi-knotoid into a labeled braidoid diagram whose closure is isotopic to the initial (multi-)knotoid. With our algorithms and defined closure, we obtain a theorem which is analogous to the classical Alexander theorem for knotoids. After this, we adapt the classical $L$-moves on braidoid and labeled braidoid diagrams that generate an extended equivalence relation on them, called the $L$-equivalence together with braidoid isotopy. e show that the $L$-equivalence provides a bijection between the set of multi-knotoids and the $L$-equivalence classes of labeled braidoid diagrams. This provides an analogous theorem to the $L$-move analogoue of the Markov theorem for braidoids that we give a proof herein. We note that it would not be possible to have such a result without the concept of the $L$-moves, since we do not have in hand an algebraic structure for braidoids. We introduce a set of elementary blocks that any braidoid is composed of, and we give the defining relations for the `multiplication' of these blocks that correspond to the braidoid isotopy moves. In this way, we show that braidoids can be encoded in terms of algebraic expressions. We end this chapter with a discussion on further questions and directions for braidoids with a small introduction to the theory of tangloids. This part of the thesis covers the results of the works with Lambropoulou. Lastly, we study topological modelings of protein chains bu utilizing the geometric interpretation we give for planar knotoids. We observe that planar knotoids provide a finer way to understand the entanglement in protein chains than the using the spherical knotoids and classical modelings utilizing closures for protein chains. We introduce the notion of bonded knotoids for modelling bonded protein chains. We study the twist insertion at bonding sites and provide a detection for sequential, pseudoknot-like and nested bonds by using knotoid invariants, such as the Turaev loop polynomial and the arrow polynomial. We end the thesis with a proposal of an algebraic encoding of polymer chains by corresponding braidoids to their knotoid models. This chapter covers the results of the work with Kauffman, Lambropoulou, Stasiak, Goundaroulis and Dorier.

Στην παρούσα διπλωματική εργασία, μελετάμε τη θεωρία των "knotoids" που εισήγαγε ο Β. Τουράεφ το 2012, εισάγουμε τη θεωρία των "braidoids" και τέλος εφαρμόζουμε τη θεωρία των κλεοειδών στη μελέτη των πρωτεϊνών.