On Skein Algebras And Sl_2(C)-Character Varieties

TL;DR

This paper explores the deep connection between skein modules of 3-manifolds and SL_2(C)-character varieties, establishing isomorphisms and developing topological methods to study these algebraic structures.

Contribution

It proves that the skein algebra over complex numbers is isomorphic to the coordinate ring of the SL_2-character variety, and develops a topological approach to character varieties.

Findings

Skein algebra over complex numbers is isomorphic to the SL_2-character variety coordinate ring.

Develops a topological framework for studying SL_2-character varieties.

Provides results on the structure of relative Kauffman bracket skein algebras.

Abstract

This paper gives insight into 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. Let R be a ring with an invertible element A. For any 3-manifold M one can assign an R-module called the Kauffman bracket skein module of M. If A^2=1 then this module has a structure of an R-algebra. We investigate this structure and, in particular, we prove that if R is the field of complex numbers then this algebra is isomorphic to the (unreduced) coordinate ring of the SL_2-character variety of pi_1(M). Using that result we develop a theory of Sl_2-character varieties by use of topological methods. We also assign to any surface a relative Kauffman bracket skein algebra. We prove several results about this non-commutative algebra. Our work should be considered in the context of the book of Brumfiel and Hilden `SL(2) Representations of Finitely Presented Groups,' Cont. Math 187. In particular we give a topological interpretation to algebraic objects considered in that book.