Double affine Hecke algebras and generalized Jones polynomials

Abstract

In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^3$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct 3-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev in \cite{RT90}. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

44pg, 3 figures, updated example polynomials, added sec. 3.4 and 5.5, minor corrections

References in corpus (1)

• A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres

Cited by in corpus (8)

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• DAHA and skein algebra on surface: double-torus knots
• Iterated torus knots and double affine Hecke algebras
• Modules over plane curve singularities in any ranks and DAHA
• On the genus two skein algebra