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- W3103912917 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." @default.
- W3103912917 created "2020-11-23" @default.
- W3103912917 creator A5015995989 @default.
- W3103912917 creator A5024075990 @default.
- W3103912917 creator A5059418519 @default.
- W3103912917 creator A5090178322 @default.
- W3103912917 date "2012-01-01" @default.
- W3103912917 modified "2023-10-18" @default.
- W3103912917 title "Volume conjecture: refined and categorified" @default.
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- W3103912917 doi "https://doi.org/10.4310/atmp.2012.v16.n6.a3" @default.
- W3103912917 hasPublicationYear "2012" @default.