𝑆𝐿_{𝑛}-character varieties as spaces of graphs

Abstract

An S L n SL_n -character of a group G G is the trace of an S L n SL_n -representation of G . G. We show that all algebraic relations between S L n SL_n -characters of G G can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space X , X, with π 1 ( X ) = G . \pi _1(X)=G. We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of S L n SL_n -representations of groups. The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of M M which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the S L 2 SL_2 -character variety of π 1 ( M ) . \pi _1(M). This paper provides a generalization of this result to all S L n SL_n -character varieties.