Multivariable Link Signatures as Casson-Type Invariants

This thesis studies link invariants of the Casson-Lin type using techniques of classical knot theory, representation theory, and gauge theory.

In the first part of the thesis, we study the multivariable Casson-Lin invariant of Bénard and Conway, defined by counting irreducible special unitary representations of the link group with prescribed meridional holonomy. Bénard and Conway calculated this invariant for two-component links with linking number one. We extend their calculation to all two-component links with non-zero linking number. To be precise, for all such links, and for holonomy parameters outside the zero set of the multivariable Alexander polynomial, we prove that this invariant equals the symmetrized Cimasoni-Florens multivariable link signature. Our proof uses a crossing change formula to reduce the computation to torus links, followed by an explicit analysis of intersection numbers in the special unitary pillowcase using Chebyshev polynomials.

In the second part of the thesis, we develop a gauge-theoretic version of the Bénard-Conway invariant. Motivated by Taubes' interpretation of the Casson invariant and by Herald's work, we define an integer-valued invariant for links of any number of components as a signed count of gauge equivalence classes of irreducible flat special unitary connections on the link exterior with prescribed meridional holonomy. For two-component links with nonzero linking number, we prove that this invariant admits a closed-form formula in terms of multivariable link signatures, and hence agrees with the Bénard-Conway invariant up to normalization. To prove this, we analyze the local structure of flat moduli spaces and establish their non-degeneracy after a small perturbation. We express the invariant as the spectral flow of a family of twisted odd signature operators, and we identify the latter with the multivariable link signature using index theory and a theorem of Toffoli. Extending these results beyond this setting would require a detailed study of bifurcations at the trivial connection.