Peter Samuelson

If M is a 3-manifold, the Kauffman bracket skein module is a vector space K_q(M) functorially associated to M that depends on a parameter q in C*. If F is a surface, then K_q(Fx [0,1]) is an algebra, and K_q(M) is a module over K_q((∂M) x [0,1]). One motivation for the definition is that if L ⊂ S³ is a knot, then the (colored) Jones polynomials J_n(L) (which are Laurent polynomials in q) can be computed from K_q(S³\setminus L).

It was shown by Frohman and Gelca that K_q(T²x [0,1]) is isomorphic to A_q^{Z₂}, the subalgebra of the quantum torus XY = q²YX which is invariant under the involution inverting X and Y. Our starting point is the observation that the category of A_q^{Z₂}-modules is equivalent to the category of modules over a simpler algebra, the crossed product A_q \rtimes Z₂. We write M_L for the image of K_q(S³\setminus L) under this equivalence. Theorem 5.2.1 gives a simple formula showing J_n(L) can be computed from M_L, and Corollary 5.3.3 shows a recursion relation for J_n(L) can be computed from M_L (if M_L is f.g. over C[X^±1]). In Chapter 6 we give an explicit description of M_L when L is the trefoil. Conjecture 4.3.4 conjectures the general structure of M_L for torus knots.

The algebra A_q \rtimes Z₂ is the t = 1 subfamily of the double affine Hecke algebra H_q,t of type A₁. In Chapter 8 we give a new skein-theoretic realization of the spherical subalgebra H⁺_q,t, and we also give a construction associating an H⁺_q,t-module M′_L(t) to each knot L. In Chapter 9 we construct algebraic deformations of the skein module M_L to a family of modules M_L(t) over H_q,t. In the case when L is the trefoil, we use these deformations to give example calculations of 2-variable polynomials J_n(q,t) that specialize to the colored Jones polynomials when t = 1.