Kauffman Bracket Skein Module for the disk sum of A times S1 and A times I.

摘要

Skein modules, as invariants of 3-manifolds, were introduced by J. Przytycki and V. Turaev in 1987 and have since become a popular area of research investigations. In particular, skein modules provide sound algebraic structures for studying 3-dimensional manifolds and knot theory in 3-manifolds. In most cases, skein modules carry an additional algebraic structure such as Hopf algebra, Lie algebra, coordinate ring of SL(2, C ) - character variety of representations of the fundamental group of a 3-manifold M3, etc. The Kauffman Bracket Skein Module (KBSM) is the most extensively studied skein module. Computing and understanding the structure of the KBSM is usually a challenging task even for 3-manifolds with simple geometric structure. In a recent paper M. Dabkowski and M. Mroczkowski computed the KBSM for F 0,3 x S1; (F 0,3 denotes the surface of genus 0 with 3 boundary components) and showed that the KBSM for F0,3 x S 1 is free. In this dissertation we outline the method of calculation for the KBSM of the manifold M3 that is obtained from the disk sum of A x S 1 and A x I, (where A is an annulus) and demonstrate how to extend the method used by Dabkowski and Mroczkowski to this manifold.