Invariants of fibred knots from moduli

Abstract

An invariant $μ_α(K)$ of fibred knots $K$ in a homology sphere is defined for each $α\in {\bold S}{\bold U}_n$ 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 $μ_α(K)$ to be the Lefschetz number of this action. The dependence of $μ_α(K)$ on $α$ is studied and formulas relating $μ_α(K)$ to $μ_β(K)$ are derived for $α,β\in ${\bold S}{\bold U}_n$.$