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In this paper we show that the normal closure of the mth power of ahalf-twist has infinite index in the mapping class group of a punctured sphere.Furthermore, in some cases we prove that the quotient of the mapping classgroup of the punctured sphere by the normal closure of a power of a half-twistcontains a free abelian subgroup. As a corollary we prove that the quotient ofthe hyperelliptic mapping class group of a surface of genus at least two by thenormal closure of the mth power of a Dehn twist has infinite order, and forsome integers m the quotient contains a free nonabelian subgroup. As a secondcorollary we recover a result of Coxeter: the normal closure of the mth powerof a half-twist in the braid group of at least five strands has infinite indexif n is at least four. Our method is to reformulate the Jones representation ofthe mapping class group of a punctured sphere, using the action of Heckealgebras on W-graphs, as introduced by Kazhdan-Lusztig.

The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere