Remarks on the faithfulness of the Jones representations

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.