Invariants of fibred knots from moduli

Abstract

An invariant ${\mu }_{\alpha }(K)$ of fibred knots $K$ in a homology sphere is defined for each $\alpha \in \mathbf{S}{\mathbf{U}}_n$ 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 $\mathbf{S}{\mathbf{U}}_n$ representations of the surface group. Restricting attention to those representations with holonomy along the longitude conjugate to $a\in \mathbf{S}{\mathbf{U}}_n,$ 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$.$

amslatex, 9 pages

References in corpus (1)

• Rationality of Moduli Spaces of Parabolic Bundles