## The Kauffman bracket Skein algebra of the punctured Torus

##### Abstract

This dissertation studies the Kauffman bracket skein algebra of the punctured torus.
The first chapter contains the historical background on the Kauffman bracket skein
algebra and its applications.
The second chapter contains the multiplication rule for the Kauffman bracket skein
algebra of the cylinder over the punctured torus. The explicit formula for the
multiplicative rule for the case of the Kauffman bracket skein algebra of the cylinder over
torus was found by Frohman and Gelca. In this work, we try to extend their result to the
torus with a puncture. The punctured torus has a multiplicative structure of the
Kauffman bracket skein algebra that is considerably more complicated than that of the
torus, and we ilustrate this with examples for which the crossing number is small.
In Chapter 3, we describe the action of the Kauffman bracket skein algebra on certain
vector spaces that arise as relative Kauffman bracket skein modules of tangles in the
torus. We analyze several particular cases, then we derive the general formula for the
action of the Kauffman bracket skein algebra on the corresponding skein modules by
using the geometric properties of the Jones-Wenzl idempotent, which is the main result
of the dissertation.
In Chapter 4, we show how the Reshetikhin-Turaev representation of the mapping
class group of the punctured torus can be computed from the representation of the
Kauffman bracket skein algebra, and based on this we derive explicit formulas for the
matrices of the generators of the mapping class group of the punctured torus.