Invariants of fibred knots from moduli

Abstract

An invariant $\mu_{\alpha}(K)$ of fibred knots $K$ in a homology sphere is defined for each $\alpha \in {\bold S}{\bold U}_n$ as follows. Since the knot is fibred, the knot complement is described by an element of the mapping class group, which induces an action on the variety of ${\bold S}{\bold U}_n$ representations of the surface group. Restricting attention to those representations with holonomy along the longitude conjugate to $a \in {\bold S}{\bold U}_n,$ one can define $\mu_{\alpha}(K)$ to be the Lefschetz number of this action. The dependence of $\mu_\alpha(K)$ on $\alpha$ is studied and formulas relating $\mu_{\alpha}(K)$ to $\mu_{\beta}(K)$ are derived for $\alpha,\beta \in ${\bold S}{\bold U}_n$.$