# Peter Samuelson

### First Position

Postdoctoral associate at the University of Toronto### Dissertation

*Kauffman Bracket Skein Modules and the Quantum Torus*

### Advisor

### Research Area

### Abstract

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}(*F**x* [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* ⊂ *S*^{3} is a knot, then the (colored) Jones polynomials *J*_{n}(*L*) (which are Laurent polynomials in *q*) can be computed from *K*_{q}(*S*^{3}\setminus *L*).

It was shown by Frohman and Gelca that *K*_{q}(*T*^{2}*x* [0,1]) is isomorphic to *A*_{q}^{Z_{2}}, the subalgebra of the quantum torus *XY* = *q*^{2}*YX* which is invariant under the involution inverting *X* and *Y*. Our starting point is the observation that the category of *A*_{q}^{Z_{2}}-modules is equivalent to the category of modules over a simpler algebra, the crossed product *A*_{q} \rtimes Z_{2}. We write *M*_{L} for the image of *K*_{q}(*S*^{3}\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 Z_{2} is the *t* = 1 subfamily of the double affine Hecke algebra *H*_{q,t} of type *A*_{1}. 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.