Kauffman Bracket Skein Modules And The Quantum Torus

Abstract

If M is a 3-manifold, the Kauffman bracket skein module is a vector space Kq (M ) functorially associated to M that depends on a parameter q ∈ C* . If F is a surface, then Kq (F x [0, 1]) is an algebra, and Kq (M ) is a module over Kq ((∂M ) x [0, 1]). One motivation for the definition is that if L [SUBSET OF] S 3 is a knot, then the (colored) Jones polynomials Jn (L) ∈ C[q ±1 ] can be computed from Kq (S 3 \ L). It was shown in [14] that Kq (T 2 x [0, 1]) ~ AZ2 , the subalgebra of the quantum =q torus XY = q 2 Y X which is invariant under the involution X [RIGHTWARDS ARROW] X [-]1 , Y [RIGHTWARDS ARROW] Y [-]1 . Our starting point is the observation that the category of AZ2 -modules is equivalent q to the category of modules over a simpler algebra, the crossed product Aq Z2 . We write ML for the image of Kq (S 3 \ L) under this equivalence. Theorem 5.2.1 gives a simple formula showing Jn (L) can be computed from ML , and Corollary 5.3.3 shows a recursion relation for Jn (L) can be computed from ML (if ML is f.g. over C[X ±1 ]). In Chapter 6 we give an explicit description of ML when L is the trefoil. Conjecture 4.3.4 conjectures the general structure of ML for torus knots. The algebra Aq Z2 is the t = 1 subfamily of the double affine Hecke algebra Hq,t of type A1 . In Chapter 8 we give a new skein-theoretic realization of the + + spherical subalgebra Hq,t , and we also give a construction associating an Hq,t - module ML (t) to each knot L. In Chapter 9 we construct algebraic deformations of the skein module ML to a family of modules ML (t) over Hq,t . In the case when L is the trefoil, we use these deformations to give example calculations of 2-variable polynomials Jn (q, t) that specialize to the colored Jones polynomials when t = 1.