Related papers: Difference equation of the colored Jones polynomia…

If a knot has the Alexander polynomial not equal to 1, then it is linear $n$-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot…

We construct knot invariants from solutions to the Yang--Baxter equation associated to appropriately generalized left/right Yetter--Drinfel'd modules over a braided Hopf algebra with an automorphism. When applied to Nichols algebras, our…

We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of…

We study coloured invariants of torus knots $T(p,p')$ (where $p,p'$ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when $p,p'\geq h^\vee$, we use the representation theory of the corresponding principal…

The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links…

In 1993 Rosso and Jones computed for every simple, complex Lie algebra g_C and every colored torus knot in S^3 the value of the corresponding U_q(g_C)-quantum invariant by using the machinery of quantum groups. In the present paper we…

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…

There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A…

This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave…

In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular, I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I…

We calculate limits of the colored Jones polynomials of the figure-eight knot and conclude that in most cases they determine the volumes and the Chern--Simons invariants of the three-manifolds obtained by Dehn surgeries along it.

Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, \Sigma(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced…

We extend a result of Thistlethwaite [17, Theorem 1(iv)] on the structure of the Jones polynomial of alternating links to the wider class of quasi-alternating links. In particular, we prove that the Jones polynomial of any prime…

We review the Reshetikhin-Turaev approach to construction of non-compact knot invariants involving R-matrices associated with infinite-dimensional representations, primarily those made from Faddeev's quantum dilogarithm. The corresponding…

We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that…

We first study superpolynomial associated to triply-graded reduced colored HOMFLY-PT homology. We propose conjectures of congruent relations and cyclotomic expansion for it. We prove conjecture of $N=1$ for torus knot case, through which we…

The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation,…

In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field…

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…

The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2) character manifold of the peripheral torus. We compute the asymptotics of the torus knot states in terms of…