Sl_n-character varieties as spaces of graphs

TL;DR

This paper introduces a topological framework for understanding SL_n-character varieties of groups through graph relations, generalizing previous work on SL_2-character varieties and link invariants.

Contribution

It demonstrates that all algebraic relations among SL_n-characters can be represented by local graph relations, providing a unified topological approach for all SL_n-character varieties.

Findings

All relations between SL_n-characters are implied by a single local graph relation.

The approach generalizes the connection between skein modules and character varieties from SL_2 to SL_n.

Provides a topological visualization of algebraic relations in SL_n-character varieties.

Abstract

An SL_n-character of a group G is the trace of an SL_n-representation of G. We show that all algebraic relations between SL_n-characters of G can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space X, with 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 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 which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the SL_2-character variety of pi_1(M). This paper provides a generalization of this result to all SL_n-character varieties.