Related papers: Remarks on the categorification of colored Jones p…
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…
The "color" in the colored Jones polynomial is an integer parameter. In this paper, a periodic pattern of the values of the colored Jones polynomial at the second and the third roots of unity is found. If we substitute -1 to the colored…
We extend the cobordism based categorification of the virtual Jones polynomial to virtual tangles. This extension is combinatorial and has semi-local properties. We use the semi-local property to prove an applications, i.e. we give a…
We show that the family of colored Jones polynomials of the closure of a braid compute weighted sums of abelianized Lefschetz numbers associated with the action of the braid on configuration spaces. The sum is over the number of…
We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials, and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves…
This paper is concerned with nanowords, a generalization of links, introduced by Turaev. It is shown that the system of bigraded homology groups is an invariant of nanowords by introducing a new notion. This paper gives two examples which…
We define a Khovanov homotopy type for $sl_2(\mathbb{C})$ colored links and quantum spin networks and derive some of its basic properties. In the case of $n$-colored B-adequate links, we show a stabilization of the homotopy types as the…
Using the Huynh and Le quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that…
The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the…
This expository essay is aimed at introducing the Jones polynomial. We will see the encapsulation of the Jones polynomial, which will involve topics in functional analysis and geometrical topology; making this essay an interdisciplinary…
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…
Biquandle brackets define invariants of classical and virtual knots and links using skein invariants of biquandle-colored knots and links. Biquandle coloring quivers categorify the biquandle counting invariant in the sense of defining…
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov…
We show that the triply graded Khovanov-Rozansky homology of knots and links over a field of positive odd characteristic $p$ descends to an invariant in the homotopy category finite-dimensional $p$-complexes. A $p$-extended differential on…
In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to…
We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of…
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…
We discuss two realizations of the colored Jones polynomials of a knot, one from an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another one from…
Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting…
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…