The Bénard–Conway invariant of two-component links

Zedan Liu and Nikolai Saveliev

Vol. 341 (2026), No. 2, 379–401

DOI: 10.2140/pjm.2026.341.379

Abstract

The Bénard–Conway invariant of links in the 3-sphere is a Casson–Lin type invariant defined by counting irreducible $\mathrm{SU}<!--FUNCTION\; APPLICATION-->(2)$-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 nonzero linking number. A key ingredient in the proof is an explicit calculation of the Bénard–Conway invariant for $(2,2\ell )$-torus links with the help of Chebyshev polynomials.

Keywords

Casson–Lin invariant, character variety, multivariable link signature

Mathematical Subject Classification

Primary: 57K10

Secondary: 57K31

Milestones

Received: 12 December 2024

Revised: 2 December 2025

Accepted: 27 December 2025

Published: 23 March 2026