Related papers: Density spectra for knots
We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of…
We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the $q$-Weyl algebra of $q$-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is…
We review recent efforts to machine learn relations between knot invariants. Because these knot invariants have meaning in physics, we explore aspects of Chern-Simons theory and higher dimensional gauge theories. The goal of this work is to…
We propose a gauge model of quantum electrodynamics (QED) and its nonabelian generalization from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants from…
This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the…
We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0…
In recent years, several families of hyperbolic knots have been shown to have both volume and $\lambda_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume…
We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of "weight systems", finding everything to be in agreement with the conjecture that "every…
We show that the link invariants derived from 3-dimensional quantum hyperbolic geometry can be defined by means of planar state sums based on link diagrams and a new family of enhanced Yang-Baxteroperators (YBO) that we compute explicitly.…
In this paper, we conjecture a connection between the $A$-polynomial of a knot in $\mathbb{S}^{3}$ and the hyperbolic volume of its exterior $\mathcal{M}_{K}$ : the knots with zero hyperbolic volume are exactly the knots with an…
We extend the Neumann's methods and give the explicit formulae for the volume and the Chern-Simons invariant for hyperbolic alternating knot orbifolds.
Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to…
We investigate the conjectural relations between the Reshetikhin-Turaev-Witten quantum SU(2) invariants and the volume of hyperbolic 3-manifolds. Given a finite set of sufficiently large positive integers, say J, we construct examples of…
A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2…
We summarize recent work on a combinatorial knot invariant called knot contact homology. We also discuss the origins of this invariant in symplectic topology, via holomorphic curves and a conormal bundle naturally associated to the knot.
We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo $4\pi^2$.
We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the M\"obius function. We also prove a conjecture about further divisibility…
We will announce some results on the values of quantum sl_2 invariants of knots and integral homology spheres. Lawrence's universal sl_2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the…
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert…
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.