The Benard-Conway invariant of two-component links

TL;DR

This paper extends the known relationship between the Benard-Conway invariant and the link signature from links with linking number one to all two-component links with non-zero linking number, using explicit calculations for torus links.

Contribution

It generalizes the connection between the Benard-Conway invariant and the link signature to all two-component links with non-zero linking number, including explicit computations for torus links.

Findings

The Benard-Conway invariant equals a symmetrized multivariable link signature for all two-component links with non-zero linking number.

Explicit calculation of the invariant for (2, 2n)-torus links using Chebyshev polynomials.

Extension of previous results from linking number one to all non-zero linking number cases.

Abstract

The Benard-Conway invariant of links in the 3-sphere is a Casson-Lin type invariant defined by counting irreducible SU(2) representations of the link group with fixed meridional traces. For two-component links with linking number one, the invariant has been shown to equal a symmetrized multivariable link signature. We extend this result to all two-component links with non-zero linking number. A key ingredient in the proof is an explicit calculation of the Benard-Conway invariant for (2, 2n)-torus links with the help of the Chebyshev polynomials.