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 ${ \bold S}{ \bold U}_n$.$
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