Volume Conjecture: Refined and Categorified

TL;DR

This paper refines and categorifies the volume conjecture by introducing a deformation parameter, leading to new algebraic curves and quantum equations that connect knot invariants, homologies, and BPS invariants.

Contribution

It proposes refined and categorified versions of the volume and AJ-conjectures, incorporating an extra parameter t and computing associated classical and quantum curves.

Findings

Derived t-deformed algebraic curves from knot homologies.

Computed quantum t-deformed curves in multiple examples.

Connected refined invariants to algebraic and quantum curves.

Abstract

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 conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation that annihilates the colored Jones polynomials and $SL(2,\C)$ Chern-Simons partition functions. We propose refinements / categorifications of both conjectures that include an extra deformation parameter $t$ and describe similar properties of homological knot invariants and refined BPS invariants. Much like their unrefined / decategorified predecessors, that correspond to $t=-1$, the new volume conjectures involve objects naturally defined on an algebraic curve $A^{ref} (x,y; t)$ obtained by a particular deformation of the A-polynomial, and its quantization $\hat A^{ref} (\hat x, \hat y; q, t)$. We compute both classical and quantum t-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.