Related papers: Jones rational coincidences
HOMFLY polynomials are one of the major knot invariants being actively studied. They are difficult to compute in the general case but can be far more easily expressed in certain specific cases. In this paper, we examine two particular…
We calculate Jones polynomials $V_L(t)$ for several families of alternating knots and links by computing the Tutte polynomials $T(G,x,y)$ for the associated graphs $G$ and then obtaining $V_L(t)$ as a special case of the Tutte polynomial.…
Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n^{th}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the…
We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This…
The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are $q$-holonomic, that is, they satisfy linear $q$-difference equations with coefficients Laurent…
A sequence of coefficients that appeared in the evaluation of a rational integral has been shown to be unimodal. An alternative proof is presented.
In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when $J_{K,0}$…
Jones and Boston conjectured that the factorization process for iterates of irreducible quadratic polynomials over finite fields is approximated by a Markov model. In this paper, we find unexpected and intricate behavior for some quadratic…
We provide a bound on the maximum degree of the Jones polynomial of any positive link with second Jones coefficient equal to $\pm 1$. This builds on the result of our previous work, in which we found such a bound for positive fibered links.
We study the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot in $S^3$. We confirm the AJ conjecture for $(r,2)$-cables of the $m$-twist knot, for all odd integers $r$ satisfying $\begin{cases}…
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof…
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 present a simple phenomenological formula which approximates the hyperbolic volume of a knot using only a single evaluation of its Jones polynomial at a root of unity. The average error is just $2.86$% on the first $1.7$ million knots,…
For a knot $K$ in $S^3$, the $sl_2$-colored Jones function $J_K(n)$ is a sequence of Laurent polynomials in the variable $t$, which is known to satisfy non-trivial linear recurrence relations. The operator corresponding to the minimal…
The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov…
We study continuity of the roots of nonmonic polynomials as a function of their coefficients using only the most elementary results from an introductory course in real analysis and the theory of single variable polynomials. Our approach…
Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and…
We address a Jordan version of Johnson theorem on (associative) algebras of quotients, namely whether a strongly nonsingular (the Jordan version of nonsingularity) has a von Neumann regular algebra of quotients. Although the answer is…
In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation.…
We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…