Volume Conjecture: Refined and Categorified

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.