Braid groups and Iwahori-Hecke algebras
Stephen Bigelow
Abstract
This article was submitted to a volume under preparation, with Benson Farb as the editor, on the topic of open problems in surface mapping class groups. The braid group Bn is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra Hn is a quotient of the braid group algebra of Bn by a quadratic relation in the standard generators. We discuss how to use Hn to define the Jones polynomial of a knot or link. We also summarize the classification of the irreducible representations of Hn. We conclude with some directions for future research that would apply mapping class group techniques to questions related to Hn.
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