A multivariable Casson-Lin type invariant

Abstract

We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\mathrm{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable signatures. We also obtain some results concerning deformations of $\mathrm{SU}(2)$ representations of link groups.

39 pages, 6 figures; v2: The statement of Theorem 1.1 has changed: our invariant is not equal to the multivariable signature but to a sum of multivariable signatures; the proof has been amended accordingly, final version, to appear in Annales de l'institut Fourier