Related papers: The Slope Conjecture for 3-String Montesinos Knots
We show that for a torus knot the SL(2;C) Chern-Simons invariants and the SL(2;C) twisted Reidemeister torsions appear in an asymptotic expansion of the colored Jones polynomial. This suggests a generalization of the volume conjecture that…
It is shown that the diameter of the boundary slope set is bounded from above by the twice of the minimal crossing number for a Montesinos knot.
It is conjectured that for each knot $K$ in $S^3$, the fundamental group of its complement surjects onto only finitely many distinct knot groups. Applying character variety theory we obtain an affirmative solution of the conjecture for a…
We discuss the higher order stabilization of the coefficients of the colored Jones polynomial. In particular, we find an expression for the second stable sequence of the colored Jones polynomial of a certain class of knots. We also…
We give an alternate expansion of the colored Jones polynomial of pretzel links which recovers the degree formula in arXiv:1807.00957. As an application, we determine the degrees of the colored Jones polynomials of a new family of 3-tangle…
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 consider the potential function of the colored Jones polynomial for a link with arbitrary colors and obtain the cone-manifold structure for the link complement. In addition, we establish a relationship between a saddle point equation and…
The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial…
The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;C) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N…
We solve the Jones conjecture, which states that the exponent sum in a minimal braid representation of a knot in S^3 is a knot invariant, by proving a generalized version of the original one. We apply contact geometry to study this problem…
The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for…
We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary…
A polynomial knot is a smooth embedding $\kappa: \real \to \real^n$ whose components are polynomials. The case $n = 3$ is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological…
We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a $q$-state Potts model defined as a planar…
We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…
The slope is an isotopy invariant of colored links with a distinguished component, initially introduced by the authors to describe an extra correction term in the computation of the signature of the splice. It appeared to be closely related…
Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an…
This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox…
The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character…
The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in…