Double affine Hecke algebras and generalized Jones polynomials (Q2816792)

The Kauffman bracket skein module \(K_q(M)\) of a 3-manifold \(M\) is a vector space depending on a complex parameter \(q\), spanned by framed links in \(M\) modulo the Kauffman bracket skein relation.NEWLINENEWLINEIt is well known that \(K_q(M)\) may be thought of as a quantum deformation of the ring of regular functions on the \(\mathrm{SL}_2(\mathbb{C})\)-character variety of \(\pi_1(M)\) [\textit{J. H. Przytycki} and \textit{A. S. Sikora}, Topology 39, No. 1, 115--148 (2000; Zbl 0958.57011)].NEWLINENEWLINEThe Kauffman bracket skein module \(K_q(S^3\setminus K)\) of the complement of a knot \(K\) in \(S^3\) admits a natural action by the Kauffman bracket skein module \(K_q(T^2\times[0,1])\) of the cylinder over the torus.NEWLINENEWLINEThe paper under review aims at introducing new parameters into the study of the skein modules by considering actions of the rank 1 double affine Hecke algebra, which is a 2-parameter deformation of an algebra loosely connected with the ring of regular functions on the \(\mathrm{SL}_2(\mathbb{C})\)-character variety of \(\mathbb{Z}^2\cong\pi_1(T^2)\).NEWLINENEWLINEThe authors conjecture that there is a natural action of a rank 1 double affine Hecke algebra on \(K_q(S^3\setminus K)\), and that there is an embedding \(K_q(S^3\setminus\text{unknot})\hookrightarrow K_q(S^3\setminus K)\) of modules over \(K_q(T^2\times[0,1])\). These conjectures are proved for several cases: the unknot, the \((2,2p+1)\)-torus knots and the figure eight knot (for general \(q\)); and the 2-bridge knots when \(q=-1\).NEWLINENEWLINEAs applications of the first conjecture, the authors define 3-variable polynomial knot invariants \(J_n(K;q,t_1,t_2)\in\mathbb{C}[q^{\pm1},t_1^{\pm1},t_2^{\pm1}]\). The conjecture also suggests that certain linear combinations with nontrivial denominators of the colored Jones polynomials of a knot are Laurent polynomials. The authors prove this fact for all knots by using the so-called cyclotomic expansion of the colored Jones polynomials. The conjecture also implies that there is an inhomogeneous recursion relation for the colored Jones polynomials of a knot.