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 $ħ$; 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 $SL(2, {\mathbb C})$ 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 $\widehat{A}^{ref} \widehat{x}, \widehat{y}; q, t)$. We compute both classical and quantum $t$-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.