Title: On Skein Algebras and Sl_2(C)-Character Varieties

Authors: Jozef H. Przytycki and Adam S. Sikora

Preprint: GWUM-1999-06 (To appear in Topology)

Abstract:
Let M be an oriented 3-manifold. For any commutative ring R with a specified invertible element A one can assign an R-module called the Kauffman bracket skein module of M. This invariant of $3$-manifolds was introduced by the first author in 1987.

This paper gives insight into broad and intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. This connection was first observed by D. Bullock in 1995.

We believe that our research, which was originated in [P-S-1] and continued in this paper, will ultimately result in a theory which will reveal some of the mysterious mutual correlations between a skein approach and an approach via methods of representation theory to 3-dimensional topology.

Such a theory is needed, for example, in order to advance the study of quantum invariants. These invariants can be defined both in terms of skein theory and representation theory (of quantum groups). A lack of good understanding of relations between these two theories gives rise to some
difficulties in studying quantum invariants.

Our work can be also considered in a context of the theory of Culler and Shalen which relates properties of the Sl_2(C)-character variety of pi_1(M), for a given 3-manifold M, with properties of incompressible surfaces in M. In this paper we give a topological interpretation of Sl_2(C)-character variety of pi_1(M). Using this interpretation one can restate some of the deep results of Culler and Shalen in a purely topological manner.