Related papers: Skein module deformations of elementary moves on l…
We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard,…
We examine computer experiments that can be performed to understand the dynamics of knots under self-repulsion. In the course of specific computer exploration we use the knot theory of rational knots and rational tangles to produce classes…
In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in $F\times S^1$ and $N\hat{\times}S^1$, where $F$ is an orientable and $N$ an unorientable surface.…
This paper introduces an algebra structure on the part of the skein module of an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in $H_1(M;\mathbb{Z}_2)$ when the value of the parameter used in the Kauffman bracket skein…
In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the…
An analytical model founded on geometric and potential energy principles for kink band deformation in laminated composite struts is presented. It is adapted from an earlier successful study for confined layered structures which was…
For each skein module we describe a homology theory which, for any three manifold recovers the skein module at its zero level. The theory measures skein-like relations among skein relations, mimicking Hilbert's theory of syzygies. We work…
It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence…
The Kauffman bracket skein module $S(M)$ of a 3-manifold $M$ is a $\mathbb{Q}(A)$-vector space spanned by links in $M$ modulo the so-called Kauffman relations. In this article, for any closed oriented surface $\Sigma$ we provide an explicit…
In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link…
We discuss different invariants of knots and links that depend on a primitive root of unity. We clarify the definitions of existing invariants with the Reshetikhin-Turaev method, present the generalization of ADO invariants to…
A $4$-move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove…
This is a PhD thesis about low dimensional topology, in particular knot thory in 3-manifolds also different from the 3-sphere, topological applications of quantum invariants, and Turaev's shadows. There is an introduction and a survey for…
We propose an unexpected twist to description of the geometry and topology of configurations of n straight lines considered as a whole 3D entity (because the lines are inseparably linked pairwise while having linking numbers 1/2 or -1/2)…
We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the…
Singular knot theory extends classical knot theory by allowing transverse double points without over/under information, together with singular Reidemeister moves of types IV and V. A central open problem in this theory is to determine the…
Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds,…
Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a…
We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where…
This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be…