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 h{stroke}; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation. © 2012 International Press.