The Torelli group and the Kauffman bracket skein module

TL;DR

This paper presents a novel embedding of the Torelli group into the completed Kauffman bracket skein algebra, providing a new approach to constructing the first Johnson homomorphism for surfaces with boundary.

Contribution

It introduces a new embedding of the Torelli group into the skein algebra, offering a fresh construction of the Johnson homomorphism for surfaces with boundary.

Findings

Embedding of Torelli group into skein algebra established

New construction of the first Johnson homomorphism provided

Enhances understanding of surface mapping class groups

Abstract

We introduce an embedding of the Torelli group of a compact connected oriented surface with non-empty connected boundary into the completed Kauffman bracket skein algebra of the surface, which gives a new construction of the first Johnson homomorphism.