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,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 Â^(ref)(xˆ,ŷ;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.

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