Related papers: The colored Jones polynomials as vortex partition …
The twisted index of 3d $\mathcal{N}=2$ gauge theories on $S^1 \times \Sigma$ has an algebro-geometric interpretation as the Witten index of an effective supersymmetric quantum mechanics. In this paper, we consider the contributions to the…
In this paper we consider an Abelian Gauge Theory in R^4 equipped with the Minkowski metric. This theory leads to a system of equations, the Klein-Gordon-Maxwell equations, which provide models for the interaction between the…
Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be…
The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume…
Let $K_{(m,p)}$ denote the family of double twist knots where $2m-1$ and $2p$ are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of…
Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the…
We prove that the N-colored Jones polynomial for the torus knot T_{s,t} satisfies the second order difference equation, which reduces to the first order difference equation for a case of T_{2,2m+1}. We show that the A-polynomial of the…
In this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored…
The amplitudes of refined Chern-Simons (CS) theory, colored by antisymmetric (or symmetric) representations, conjecturally generate the Lambda^r- (or S^r-) colored triply graded homology of (n,m) torus knots. This paper is devoted to the…
We consider $\mathcal{N}=2$ supersymmetric pure gauge theories on toric K\"ahler manifolds, with particular emphasis on $\mathbb{CP}^2$. By choosing a vector generating a $U(1)$ action inside the torus of the manifold, we construct…
We show the $n$ colored Jones polynomials of a highly twisted link approach the Kauffman bracket of an $n$ colored skein element. This is in the sense that the corresponding categorifications of the colored Jones polynomials approach the…
We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.
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…
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the…
We formulate a conjecture about the structure of `upper lines' in the expansion of the colored Jones polynomial of a knot in powers of (q-1). The Melvin-Morton conjecture states that the bottom line in this expansion is equal to the inverse…
It has been a long-standing problem how to relate Chern-Simons theory to the quantum groups. In this paper we recover the classical $r$-matrix directly from a 3-dimensional Chern-Simons theory with boundary conditions, thus creating a…
This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes…
Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially.…
We study supersymmetric gauge theories in five dimensions, using their relation to the K-theory of the moduli spaces of torsion free sheaves. In the spirit of the BPS/CFT correspondence the partition function and the expectation values of…
Coset constructions in the framework of Chern-Simons topological gauge theories are studied. Two examples are considered: models of the types ${U(1)_p\times U(1)_q\over U(1)_{p+q}}\cong U(1)_{pq(p+q)}$ with $p$ and $q$ coprime integers, and…