2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75085This 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 B_n is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra H_n is a quotient of the braid group algebra of B_n by a quadratic relation in the standard generators. We discuss how to use H_n to define the Jones polynomial of a knot or link. We also summarize the classification of the irreducible representations of H_n. We conclude with some directions for future research that would apply mapping class group techniques to questions related to H_n.15 pages, 4 figuresGeometric Topology20F36 (Primary) 20C08, 57M27 (Secondary)text