A Casson-Lin type invariant for links

TL;DR

This paper introduces an integer-valued invariant for two-component links in three-dimensional space, linking it to the classical linking number through representations of the link group.

Contribution

It generalizes Lin's knot invariant to links, connecting projective SU(2) representations with the linking number in a novel way.

Findings

Invariant equals the linking number up to sign

Generalizes Lin's knot invariant to links

Provides a new perspective on link invariants

Abstract

We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking number of the link. Our construction generalizes that of X.-S. Lin who defined a similar invariant for knots in S^3; his invariant equals half the knot signature.