Related papers: Volume Conjecture: Refined and Categorified
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 the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then…
In this article, we give a rough, and so not complete yet, proof of Kashaev's conjecture, that is, the volume conjecture for hyperbolic knots, where the hyperbolicity equations associated to knot diagrams appear as the stationary phase…
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one…
We use recent progress on Chern-Simons gauge theory in three dimensions [18] to give explicit, closed form formulas for the star product on some functions on the affine space ${\mathcal A}(\Sigma)$ of (smooth) connections on the trivialized…
A conjecture of Mumford predicts a complete set of relations between the generators of the cohomology ring of the moduli space of rank 2 semi-stable sheaves with fixed odd degree determinant on a smooth, projective curve of genus at least…
Solutions to both the diffeomorphism and the hamiltonian constraint of quantum gravity have been found in the loop representation, which is based on Ashtekar's new variables. While the diffeomorphism constraint is easily solved by…
Knot polynomials colored with symmetric representations of $SL_q(N)$ satisfy difference equations as functions of representation parameter, which look like quantization of classical ${\cal A}$-polynomials. However, they are quite difficult…
Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, C_q[X_G(torus)], of the coordinate ring of the G-character variety of the…
It has been proposed that quantum complexity is dual to the volume of the extremal surface, the action of the Wheeler-DeWitt patch, and the spacetime volume of the patch. Recently, a generalized volume-complexity observable was formulated…
In the first of these two lectures, I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge…
We explore an asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.
We compare eight versions of finite-dimensional categorifications of the colored Jones polynomial and show that they yield isomorphic results over a field of characteristic zero. As an application, we verify a physics-motivated conjectural…
The A-polynomial is a knot invariant related to the space of $SL_2(\mathbb{C})$ representations of the knot group. In this paper our interests lies in the logarithmic Gauss map of the A-polynomial. We develop a homological point of view on…
The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…
X.-S. Lin and Z. Wang recently made a conjecture concerning the integrality of the Taylor coefficients of the averaged Jones polynomial of algebraically split links. This question is related to a conjectural integrality result for the…
We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley's ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for…
We define a renormalized volume for a region in an asymptotically hyperbolic Einstein manifold that is bounded by a Graham-Witten minimal surface and the conformal infinity. We prove a Gauss-Bonnet theorem for the renormalized volume, and…
We conjecture a relation between generalized quiver partition functions and generating functions for symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincar\'e polynomials of a knot $K$. We interpret the…
A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that…