Related papers: The Large-Color Expansion Derived from the Univers…

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY…

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect…

We study the asymptotic behavior, as $N$ tends to infinity, of the $N$-dimensional colored Jones polynomial of the figure-eight knot, evaluated at $\exp(\xi/N)$ for a complex parameter $\xi$ with $0<\mathrm{Im}\xi<\pi/2$. We prove that if…

The colored Jones polynomial is a knot invariant that plays a central role in low dimensional topology. We give a simple and an efficient algorithm to compute the colored Jones polynomial of any knot. Our algorithm utilizes the walks along…

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 monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be…

The $2$-decomposition for ribbon graphs was introduced in [Annals of Combinatorics 15 (2011), pp 675-706]. We extend this result to half-edged ribbon graphs and to rank $D$-weakly colored graphs [SIGMA 12 (2016), 030], generalizing…

Colored knot polynomials possess a peculiar Z-expansion in certain combinations of differentials, which depends on the representation. The coefficients of this expansion are functions of the three variables (A,q,t) and can be considered as…

This is the first article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki, we obtain an asymptotic…

In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of…

We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the $q$-Weyl algebra of $q$-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is…

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we…

We study the volume conjecture of the colored Jones invariants with sequences of colors corresponding to the deformation of the hyperbolic structure of a link complement. In particular, we investigate certain limits of the colored Jones…

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum…

Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge…

We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.

The Bott-Cattaneo-Rossi invariant $(Z_k)_{k\in \mathbb N\setminus\{0,1\}}$ is an invariant of long knots $\mathbb R^n\hookrightarrow\mathbb R^{n+2}$ for odd $n$, which reads as a combination of integrals over configuration spaces. In this…

We generalise the finite biquandle colouring invariant to a polynomial invariant based on labelling a knot diagram with a finite birack that reduces to the biquandle colouring invariant in that case. The polynomial is an invariant of a…

Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern-Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in…