  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.
