Related papers: Integer Knot Invariants: Inequalities, Computation…

We explicitly show that the new polynomial invariants for knots, upto nine crossings, agree with the Ooguri-Vafa conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold.

According to the idea of Ozsv\'ath, Stipsicz and Szab\'o, we define the knot invariant $\Upsilon$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that…

As a generalization of the classical knots, knotoids deal with the open ended knot diagrams in a surface. In recent years, many polynomial invariants for knotoids have appeared, such as the bracket polynomial, the index polynomial and the…

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects…

Given a virtual knot $K$, we construct a group $VG_K$ called the virtual knot group, and we use the elementary ideals of $VG_K$ to define invariants of $K$ called the virtual Alexander invariants. For instance, associated to the $k=0$ ideal…

We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The…

Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state. Over the past decades, these invariants have come to play a central role in describing matter,…

This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying…

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister…

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…

We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined…

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants…

We give empirical evidence that the UV-divergences of a renormalizable field theory are knot invariants.

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify…

In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon…

We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases. We also argue that, for $k \geq 18$, most graphs of order $k$ are…

In [8], K. Kaur, S. Kamada et al. posed a problem of finding a virtual knot, if exists, with an unknotting index (n,m), where (n,m) is a pair of non-negative integers. In this paper, we address this question by providing infinite families…

We develop the study of the twelve intersection polynomials of long virtual knots, previously introduced in our preceding paper. We define two geometric invariants, the $1$- and $2$-supporting genera, using two distinct surface…

We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of…

We define and study Vassiliev invariants for (long) Morse knots. It is shown that there are Vassiliev invariants which can distinguish some topologically equivalent Morse knots. In particular, there is an invariant of order 3 for Morse…