The homological content of the Jones representations at \(q= -1\) (Q2833316)

The \textit{Jones representations} of braid groups proposed by \textit{V. F. R. Jones} [in: Geometric methods in operator algebras, Proc. US-Jap. Semin., Kyoto/Jap. 1983, Pitman Res. Notes Math. Ser. 123, 242--273 (1986; Zbl 0659.46054)] which lead to the celebrated \textit{Jones polynomial}, were discovered to be related to a homological invariant in [\textit{Y. Kasahara}, Algebr. Geom. Topol. 1, 39--55 (2001; Zbl 0964.57016)]. The present paper generalizes this discovery to such an extent that the representations evaluated at \(q=-1\) are related to an action on the homology of a branched double cover of the punctured disk; explicit isomorphisms are constructed, which are of geometric nature. Then the result is used to partially prove the AMU Conjecture, which was presented in [\textit{J. E. Andersen}, \textit{G. Masbaum} and \textit{K. Ueno}, Math. Proc. Camb. Philos. Soc. 141, No. 3, 477--488 (2006; Zbl 1110.57009)], claiming the infiniteness of orders of pseudo-Anosov mapping classes. This adds to various examples of applying TQFT to classical topology.NEWLINENEWLINEIt occurs many times that classical invariants are incarnations of special values of quantum invariants. Recall that the Burau representation \(\rho_4\) of \(B_4\) is equivalent to a Jones representation, and this is the reason for the equivalence of the faithfulness of \(\rho_4\) and the Jones polynomial detecting the unknot (both are still open). More relations of this kind can be expected to be revealed.