The homological content of the Jones representations at q = −1

Abstract

We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at [Formula: see text], are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of [Formula: see text], Math. Proc. Cambridge Philos. Soc. 141(3) (2006) 477–488] by extending their original argument for the sphere with four marked points to our more general case.