In this thesis, we study alternating virtual knots. We show the Alexander polynomial of an almost classical alternating knot is alternating. We give a characterization theorem for alternating knots in terms of Goeritz matrices. We prove any reduced alternating diagram has minimal genus, and use this to prove the frst Tait Conjecture for virtual knots, namely any reduced diagram of an alternating virtual knot has minimal crossing number. / Thesis / Doctor of Philosophy (PhD)

The notion of finite type invariants of virtual knots, introduced by Goussarov, Polyak and Viro, leads to the study of the space of diagrams with directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures' defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T. In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces of arrow diagrams up to degree 4, and our results suggest that in degree 4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras.

The notion of finite type invariants of virtual knots, introduced by Goussarov, Polyak and Viro, leads to the study of the space of diagrams with directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures' defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T. In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces of arrow diagrams up to degree 4, and our results suggest that in degree 4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras.

In this thesis, we show that every periodic virtual knot can be realized as the closure of a periodic virtual braid. If K is a q-periodic virtual knot with quotient K_*, then the knot group G_{K_*} is a quotient of G_K and we derive an explicit q-symmetric Wirtinger presentation for G_K, whose quotient is a Wirtinger presentation for G_{K_*}. When K is an almost classical knot and q=p^r, a prime power, we show that K_* is also almost classical, and we establish a Murasugi-like congruence relating their Alexander polynomials modulo p. This result is applied to the problem of determining the possible periods of a virtual knot $K$. For example, if K is an almost classical knot with nontrivial Alexander polynomial, our result shows that K can be p-periodic for only finitely many primes p. Using parity and Manturov projection, we are able to apply the result and derive conditions that a general q-periodic virtual knot must satisfy. The thesis includes a table of almost classical knots up to 6 crossings, their Alexander polynomials, and all known and excluded periods. / Thesis / Doctor of Philosophy (PhD)

This thesis will take a look at a branch of topology called knot theory. We will first look at what started the study of this field, classical knot theory. Knot invariants such as the Bracket polynomial and the Jones polynomial will be introduced and studied. We will then explore racks and biracks along with the axioms obtained from the Reidemeister moves. We will then move on to generalize classical knot theory to what is now known as virtual knot theory which was first introduced by Louis Kauffman. Finally, we take a look at a newer aspect of knot theory, twisted virtual knot theory and we defined new link invariants for twisted virtual biracks.

<p>This thesis is an introduction to virtual knots and the forbidden moves, and the closely related classes of welded and unwelded knots. Extensions of the Jones polynomial and the knot group to the various knot types are considered. We also examine the operation of connected sum for virtual and welded knots, and we review the proof that every virtual knot can be untied using the forbidden moves.</p> / Master of Science (MSc)

Le but de cette thèse est de fournir une caractérisation topologique de tresses virtuelles. Les tresses virtuelles sont des classes d’équivalence de diagrammes de type tresses tracés sur le plan. La relation d’équivalence est générée par l’isotopie, les mouvements de Reidemeister et les mouvements de Reidemeister virtuels. L’ensemble des tresses virtuelles est munie d’une opération de groupe. On parlera alors du groupe de tresses virtuelles. Dans le Chapitre 1, nous introduisons les notions de base de la théorie de noeuds virtuels, nous évoquons certains propriétés du groupe tresses virtuelles, et des liens qu’il a avec le groupe de tresses classiques. Dans le Chapitre 2, nous introduisons la notion de diagramme de Gauss tressé (ou diagramme de Gauss horizontal), et on démontre qu’il s’agit là d’une bonne réinterprétation combinatoire pour les tresses virtuelles. On généralise en particulier certains résultats connus en théorie de noeuds virtuels. Un application est de retrouver la présentation classique du groupe de tresses virtuelles pures à l’aide des diagrammes de Gauss tressés. Dans le Chapitre 3, on introduit les tresses abstraites et on montre qu’elles sont en correspondance bijective avec les tresses virtuelles. Les tresses abstraites sont des classes d’équivalence des diagrammes de type tresses tracés sur une surface orientable avec deux composantes de bord. La relation d’équivalence est générée par l’isotopie, la compatibilité, la stabilité et les mouvements de Reidemeister. La compatibilité est la relation d’équivalence générée par les difféomorphismes préservant l’orientation. La stabilité est la relation d’équivalence générée par l’addition ou la suppression d’anses à la surface, dans le complémentaire du diagramme. Dans le Chapitre 4, on démontre que tout tresse abstraite admets une unique représentant de genre minimal, à compatibilité et mouvements de Reidemeister prés. En particulier, les tresses classiques se plongent dans les tresses abstraites. / The purpose of this thesis is to give a topological characterization of virtual braids. Virtual braids are equivalence classes of planar braid-like diagrams identified up to isotopy, Reidemeister and virtual Reidemeister moves. The set of virtual braids admits a group structure and is called the virtual braid group. In Chapter 1 we present a general introduction to the theory of virtual knots, and we discuss some properties of virtual braids and their relations with classical braids. In Chapter 2 we introduce braid-Gauss dia- grams, and we prove that they are a good combinatorial reinterpretation of virtual braids. In particular this generalizes some results known in virtual knot theory. As an application, we use braid-Gauss diagrams to recover a well known presentation of the pure virtual braid group. In Chapter 3 we introduce abstract braids and we prove that they are in a bijective cor- respondence with virtual braids. Abstract braids are equivalence classes of braid-like diagrams on an orientable surface with two boundary components. The equivalence relation is generated by isotopy, compatibility, stability and Reidemeister moves. Compatibility is the equivalence relation generated by orientation preserving diffeomorphisms. Stability is the equivalence relation generated by adding handles to or deleting handles from the surface in the complement of the braid-like diagram. In Chapter 4 we prove that for any abstract braid, there is a unique representative of minimal genus, up to compatibility and Reidemeister equivalence. In particular this implies that classical braids embed in abstract braids.