Remarks on the faithfulness of the Jones representations

TL;DR

This paper examines the faithfulness of Jones' linear representations of the mapping class group of an n-punctured sphere, linking it to Iwahori-Hecke algebra representations and the Burau representation, with specific results for n=6.

Contribution

It establishes the equivalence of faithfulness between Jones' representations and related Iwahori-Hecke algebra representations, and provides new restrictions on the kernel for n=6.

Findings

Faithfulness of Jones representations is equivalent to that of related Iwahori-Hecke algebra representations.

For n=6, the kernel of the representation is further restricted using previous results.

A relation between the Jones representation and the Burau representation of degree 4 is identified.

Abstract

We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the case of n=6, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4.