Virtual Knot Diagrams and the Witten-Reshetikhin-Turaev Invariant

Let $n$ be a positive integer. The aim of this paper is to study two local moves $V(n)$ and $V^{n}$ on welded links, which are generalizations of the crossing virtualization. We show that the $V(n)$-move is an unknotting operation on welded knots for any $n$, and give a classification of welded links up to $V(n)$-moves. On the other hand, we give a necessary condition for which two welded links are equivalent up to $V^{n}$-moves. This leads to show that the $V^{n}$-move is no...

In [13], K. Kaur, S. Kamada et al. posed a problem of finding a virtual knot, if exists, with an unknotting index (n,m), for any pair of non-negative integers (n,m). In this paper, we address this question by providing infinite families of virtual knots with unknotting index (n,m), for a given pair of non-negative integers (n,m). Additionally, we extend our result for virtual links also.

We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of ...

This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the polynomial takes values in the module generated by these reduced graphs over the ring Q[A,A^{-1}]. The paper is relatively self-contained, with background informat...

This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way it arises first in a graph-theoretic setting in relation to generalizing the Penrose evaluation for colorings of planar trivalent graphs to all trivalent graphs, and later by its uses in a virtual knot theory. As a consequence, the paper be...

By work of W. Thurston, knots and links in the 3-sphere are known to either be torus links, or to contain an essential torus in their complement, or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the...

A virtual knot, which is one of generalizations of knots in $\mathbb{R}^{3}$ (or $S^{3}$), is, roughly speaking, an embedded circle in thickened surface $S_{g} \times I$. In this paper we will discuss about knots in 3 dimensional $S_{g} \times S^{1}$. We introduce basic notions for knots in $S_{g} \times S^{1}$, for example, diagrams, moves for diagrams and so on. For knots in $S_{g} \times S^{1}$ technically we lose over/under information, but we have information "how many t...

This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and as a guide to some of the more advanced material. Our aim is to lead the reader to understanding by means of pictures and calculations, and for this reason we often prefer to convey the idea of the proof on an instructive example rather th...

The paper has been withdrawn by the author, due to a critical error stemming from the defined template.

We introduce a local deformation called the virtualized $\Delta$-move for virtual knots and links. We prove that the virtualized $\Delta$-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient condition for two virtual links to be related by a finite sequence of virtualized $\Delta$-moves.