Related papers: Jones polynomials of torus knots via DAHA
We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of…
We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular…
We study the structural properties of colored Kauffman homologies of knots. Quadruple-gradings play an essential role in revealing the differential structure of colored Kauffman homology. Using the differential structure, the Kauffman…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollob\'as and Riordan, we introduce a…
We construct 3D $\mathcal{N}=2$ abelian gauge theories on $\mathbb{S}^2 \times \mathbb{S}^1$ labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones…
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus…
Three dimensional SU(2) Chern-Simons theory has been studied as a topological field theory to provide a field theoretic description of knots and links in three dimensions. A systematic method has been developed to obtain the link-invariants…
In 2012, Cohen, Dasbach, and Russell presented an algorithm to construct a weighted adjacency matrix for a given knot diagram. In the case of pretzel knots, it is shown that after evaluation, the determinant of the matrix recovers the Jones…
The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and…
We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special…
We establish the duality between the torus knot superpolynomials or the Poincar\'e polynomials of the Khovanov homology and particular condensates in $\Omega$-deformed 5D supersymmetric QED compactified on a circle with 5d Chern-Simons(CS)…
We propose a new gauge theory of quantum electrodynamics (QED) and quantum chromodynamics (QCD) from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants…
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and…
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…
In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern--Simons invaraint and Reidemeister torsion.
This article contains general formulas for Tutte and Jones polynomials for families of knots and links given in Conway notation and "portraits of families"-- plots of zeroes of their corresponding Jones polynomials.
The expectation value of Wilson loop operators in three-dimensional SO(N) Chern-Simons gauge theory gives a known knot invariant: the Kauffman polynomial. Here this result is derived, at the first order, via a simple variational method.…
We show that for a twist knot, the A-polynomial can be obtained from recurrences for the summand in Masbaum's formula of the colored Jones polynomial. Our result supports the AJ conjecture due to S.Garoufalidis.
Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory…