Peter Samuelson

Ph.D. (2012) Cornell University

First Position

Postdoctoral associate at the University of Toronto


Kauffman Bracket Skein Modules and the Quantum Torus


Research Area

differential operators on algebraic varieties


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 in C*. If F is a surface, then Kq(Fx [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 LS3 is a knot, then the (colored) Jones polynomials Jn(L) (which are Laurent polynomials in q) can be computed from Kq(S3\setminus L).

It was shown by Frohman and Gelca that Kq(T2x [0,1]) is isomorphic to Aq^{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 Aq^{Z2}-modules is equivalent to the category of modules over a simpler algebra, the crossed product Aq \rtimes Z2. We write ML for the image of Kq(S3\setminus 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 \rtimes 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 H+q,t, and we also give a construction associating an H+q,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.