A Casson–Lin type invariant for links

Eric Harper and Nikolai Saveliev

Vol. 248 (2010), No. 1, 139–154

DOI: 10.2140/pjm.2010.248.139

Abstract

In 1992, Xiao-Song Lin constructed an invariant h(K) of knots K ⊂ S³ via a signed count of conjugacy classes of irreducible SU(2) representations of π₁(S³ −K) with trace-free meridians. Lin showed that h(K) equals one half times the knot signature of K. Using methods similar to Lin’s, we construct an invariant h(L) of two-component links L ⊂ S³. Our invariant is a signed count of conjugacy classes of projective SU(2) representations of π₁(S³ − L) with a fixed 2-cocycle and corresponding nontrivial w₂. We show that h(L) is, up to a sign, the linking number of L.

Keywords

link group, braid, projective representation

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M05

Milestones

Received: 20 July 2009

Accepted: 8 October 2009

Published: 1 October 2010