Double affine Hecke algebras and generalized Jones polynomials

Abstract

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 ⊂ 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 = ±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 [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.

Affine cubic surfaces and character varieties of knots

“…Remark 2.6. Comparing our notation to [BS16], their (T 0 , T ∨ 0 , T 1 , T ∨ 1 ) are our (T 2 , T 1 , T 3 , T 4 ), and their (t 1 , t 2 , t 3 , t 4 ) are our (t 2 , t 1 , t 3 , t 4 ).…”

“…For general q, this conjecture involves a quantization of the character variety of the knot complement. It seems generally agreed (at the moment) that the quantization (or q-deformation) of character varieties of knots requires the use of topological tools, such as the Kaufmann bracket skein module construction which was used in [BS16]. By contrast, the Hecke (or Dunkl) deformations that we study in the present paper (for q = −1) depend only on the knot group and may be performed purely algebraically (using the Brumfiel-Hilden algebra).…”

“…This implies that the skein module Sk q (S 3 \ K) of a knot complement is a module over SH q,1,1,1,1 . Based on explicit computations in some examples, the following conjecture was proposed in [BS16].…”

“…Conjecture 1.1 ( [BS16]). The double affine Hecke algebra SH q,t1,t2,1,1 acts canonically on the Kauffman bracket skein module Sk q (S 3 \ K) of the complement of a knot K ⊂ S 3 .…”

DAHA and skein algebra of surfaces: double-torus knots

“…It is known that the DAHA of C ∨ C 1 -type represents a quantization of the a ne cubic surface which is the character variety of a 4-punctured sphere [41], while the DAHA of A 1 -type is related to the character variety of a once-punctured torus. Based on the fact [9,42] that the coordinate ring of the character varieties is a specialization of the Kau man bracket skein algebra, discussed also is a relationship with the skein algebra on the 4-punctured sphere and the once-punctured torus [6,7].…”