Related papers: Knot invariants for rail knotoids
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.
Twisted torus knots are torus knots with some full twists added along some number of adjacent strands. There are infinitely many known examples of twisted torus knots which are actually torus knots. We give eight more infinite families of…
For an oriented knot $K$, we construct a functor from the category of pointed quandles to the category of quandles in three different ways. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariant of…
This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from…
In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalized knotoids to allow arbitrarily…
We construct knot invariants from solutions to the Yang--Baxter equation associated to appropriately generalized left/right Yetter--Drinfel'd modules over a braided Hopf algebra with an automorphism. When applied to Nichols algebras, our…
I present a summary of the recent progress made in field and string theory which has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be described in…
We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the…
We define a local move for knots and links called the {\em one-two-way pass-move}, abbreviated briefly as the {\em $1$-$2$-move}. The $1$-$2$-move is motivated from the pass-move and the $\#$-move, and it is a hybrid of them. We show that…
This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the…
We compute the involutive knot invariants for pretzel knots of the form P(-2,m,n) for m and n odd and greater than or equal to 3.
In this report, I will start by first giving a brief introduction on knots to build some intuition before beginning the more rigorous review in the Literature Review section. There, I will define knot equivalence, the Jones polynomial…
We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…
We define a knot invariant and a 2-knot invariant from any finite categorical group. We calculate an explicit example for the Spun Trefoil.
In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to…
We produce a facial state sum on plane diagrams of a knot or a link which admits an invariant specialization under Polyak's recent set of generating of 4 Reidemeister moves. Thus an isotopy invariant of framed links is obtained. Each state…
We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.
We discuss the existence of Galois relations obeyed by certain link invariants. Some of these relations have recently been identified and exploited within the context of CFT and Lie/Kac-Moody representation theory. These relations should…
In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2$\times$2 matrices with entries in a possibly non-commutative ring, for example the quaternions.…
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…