Related papers: New presentations of a link and virtual link
Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating…
We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of…
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up…
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 talk we will discuss about knots in 3 dimensional $S_{g}…
We prove that certain problems naturally arising in knot theory are NP--hard or NP--complete. These are the problems of obtaining one diagram from another one of a link in a bounded number of Reidemeister moves, determining whether a link…
A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by R. Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots.…
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via…
In this paper, we study a geometric/topological measure of knots and links called the nullification number. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, one can intuitively regard…
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…
We introduce a framework to analyze knots and links in an unmarked solid torus. We discuss invariants that detect when such links are equivalent under an ambient homeomorphism, and show that the multivariable Alexander polynomial is such in…
An $n$-crossing projection of a link $L$ is a projection of $L$ onto a plane such that $n$ points on $L$ are superimposed on top of each other at every crossing. We prove that for all $k \in \mathbb{N}$ and all links $L$, the inequality…
We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather…
We use the rational Witt class of a knot in the 3-sphere as a tool for addressing questions about its unknotting number. We apply these tools to several low crossing knots (151 knots with 11 crossing and 100 knots with 12 crossings) and to…
Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking…
In this paper we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy.…
We construct geometrically two universal link invariants: universal ADO invariant and universal Jones invariant, as limits of invariants given by graded intersections in configuration spaces. More specifically, for a fixed level $\mathscr…
We calculate Jones polynomials $V(H_r,t)$ for a family of alternating knots and links $H_r$ with arbitrarily many crossings $r$, by computing the Tutte polynomials $T(G_+(H_r),x,y)$ for the associated graphs $G_+(H_r)$ and evaluating these…
Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two…
We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $\Delta_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied…
F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a…