Related papers: Evolutionary Khovanov homology
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded…
We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the…
The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other…
Building on work by Alishahi-Dowlin, we extract a new knot invariant $\lambda \ge 0$ from universal Khovanov homology. While $\lambda$ is a lower bound for the unknotting number, in fact more is true: $\lambda$ is a lower bound for the…
This is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects.
We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of…
Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that…
This paper studies knots in three dimensional projective space. Our technique is to associate a virtual link to a link in projective space so that equivalent projective links go to equivalent virtual links (modulo a special flype move). We…
Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze…
In this paper we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing…
Khovanov homology is a powerful link invariant: a categorification of the Jones polynomial that enjoys a rich and beautiful algebraic structure. This homology theory has been extensively studied and it has become an ubiquitous topic in…
The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases…
A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and…
Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot.…
Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally…
We consider the probability of knotting in equilateral random polygons in Euclidean 3-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal…
The aims of this letter are three-fold: First is to show that nonlinear generalizations of electrodynamics support various types of knotted solutions in vacuum. The solutions are universal in the sense that they do not depend on the…
A knot theory for two-dimensional square lattice is proposed, which sheds light on design of new two-dimensional material with high topological numbers. We consider a two-band model, focusing on the Hall conductance {\sigma}xy = e^2/hbar*P,…
In this paper, we introduce a new type of relation between knots called the descendant relation. One knot $H$ is a descendant of another knot $K$ if $H$ can be obtained from a minimal crossing diagram of $K$ by some number of crossing…
Two natural generalizations of knot theory are the study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spatial graphs.