arXiv research

New geometric invariant from disc intersections captures all coloured Jones polynomials.

Study shows colored Jones invariants limit to link volumes.

A new knot invariant uses permutations to extend Jones polynomials.

Globalizes Jones and Alexander polynomials using topological intersections.

Constructs universal link invariants from intersections in configuration spaces.

The article calculates a multiplying factor to convert rational Vassiliev invariants to integer-valued ones.

Expands Jones polynomial for Legendrian knots with categorification.

In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the…

We find approximations by Vassiliev invariants for the coefficients of the Jones polynomial and all specializations of the HOMFLY and Kauffman polynomials. Consequently, we obtain approximations of some other link invariants arising from the homology of branched covers of links.

Unified ADO and colored Jones polynomials for knots.

Paper explores the Jones polynomial and its impact on knot theory and related fields.

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…

We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of link…

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

Study shows quantum modularity in figure-eight knot's colored Jones polynomial.

Study on the growth of colored Jones polynomial for figure-eight knot cables.

Characters from logarithmic VOAs linked to torus link invariants.

New methods reveal colored Jones polynomials from quantum R-matrices and knot invariants.

New quantum knot invariants derived from Verma modules.

In this work, we give a formula for the logarithmic invariant of knots in terms of certain derivatives of the colored Jones invariant. This invariant is related to the logarithmic conformal field theory, and was defined by using the centers in the radical of the restricted quantum group at root of unity. A relation bet…

Quantum invariants are explained as intersections in configuration spaces.

The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural numb…

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…

Categorifies Jones polynomial using Lie theory.

This paper investigates the equivalence between Yamada polynomial and Jones polynomial of associated links for brunnian θ-curves.

Study uses big data to analyze quantum invariants.

Study knot invariants to deduce Hopf invariant and propose a slope conjecture.

The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's $G_2$ quantum invariants. Our method consists in the study of Mark…

The paper connects ADO polynomials to Vassiliev invariants for knots.

The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of `the categorification of the Jones polynomial'. For the same low cost we also provide some computations, including one that shows that Khovanov's…

New link homologies categorify Jones polynomial at odd prime powers.

In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial, the Kuperberg bracket, and the normalised arrow polynomial.

We generalize the colored Jones polynomial to $4$-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.

We show examples of knots with the same polynomial invariants and hyperbolic volumes, with variously coinciding 2-cable polynomials and colored Jones polynomials, which are not mutants.

Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.

We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There i…

Paper extends Cohen's method to compute Jones polynomial for certain braid subfamilies.

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 $4π^2$.

We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjectu…

Study on quantum invariants of twist knots at specific roots of unity.

Paper proves knots satisfy a conjecture using Jones polynomial.

Study on quantum invariants of twist knots at specific roots of unity.

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distin…

Study on colored Jones polynomial of figure-eight knot for complex parameters.

Unified quantum invariants via intersections of embedded Lagrangians.

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…

Machine learning maps knots to embeddings, revealing topological invariants.

Proof of isotopy invariance in Khovanov link homology.