arXiv research

Paper connects AJ conjecture and colored Jones polynomial potential function.

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation …

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones fun…

The Jones polynomial of a knot in 3-space is a Laurent polynomial in $q$, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing defin…

Formula for arborescent link tails using theta functions.

Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative re…

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element o…

Study on colored Jones polynomial and link complements.

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false thet…

We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the …

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…

I show various calculations of the limit of the colored Jones function for the figure-eight knot and confirm R. Kashaev's conjecture in this case.

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 polynomials in $q$ and $q^n$. We show from first principles that $q$-holonomic sequence…

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 graph with weighted edges that corresponds to the knot. For any integer $q$, we cast…

Researchers compute and predict knot volumes using colored Jones polynomials.

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and…

Unified ADO and colored Jones polynomials for knots.

3D gauge theories link knot polynomials to vortex partition functions.

We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerbo…

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the $n$th colored Jones polynomial at $e^{\a/n}$, when $\a$ is a fixed complex number and $n$ tends to infinity. We analy…

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 explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …

New knot models analyze local entanglement for robust curve analysis.

A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a …

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyc…

Novel Jones polynomial for open curves in 3D space.

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model…

We formulate a conjecture (already proven by A. Kricker) about the structure of Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent diagrams. We conjecture that these functions are rational functions of the exponentials …

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathf…

Preprint proves quantum coideal Schur-Weyl duality and generalizes Jones-Wenzl projectors.

Proves volume conjecture for twist knots using complex analysis.

The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of SL(2). It was recently shown by TTQ Le and the author that the colored Jones function of a knot is q-holonomic, ie, …

New method measures entanglement of open and closed curves.

Jones slopes detect figure eight knot, and characterize alternating knots.

New method proves Jones Polynomial's connect sum property.

We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollar…

Jones polynomials derived from K-theory of a cluster algebra.

Upper bound on Jones polynomials density modulo primes.

Survey on categorifying Jones polynomial.

New formula recovers degree of colored Jones polynomials for pretzel knots.

Jones constructs knots from Thompson group elements.

New bound on Jones polynomial for specific positive links.

This paper will be an exposition of the Kauffman bracket polynomial model of the Jones polynomial, tangle methods for computing the Jones polynomial, and the use of these methods to produce non-trivial links that cannot be detected by the Jones polynomial.

Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations

Jones polynomial coincidences explored for rational knots.

Researchers lift knot coloring polynomial to Habiro ring.

Categorifies Jones polynomial using Lie theory.

We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.

New proof limits Jones polynomial values for quasi-alternating links.