On a theorem of Murasugi

Cameron Gordon and Richard A. Litherland

Vol. 82 (1979), No. 1, 69–74

DOI: 10.2140/pjm.1979.82.69

Abstract

1. Let l = k₁ ∪ k₂ be a 2-component link in S³, with k₂ unknotted. The 2-fold cover of S³ branched over k₂ is again S³; let k₁⁽²⁾ be the inverse image of k₁, and suppose that k₁⁽²⁾ is connected. How are the signatures σ(k₁), σ(k₁⁽²⁾) of the knots k₁ and k₁⁽²⁾ related? This question was considered (from a slightly different point of view) by Murasugi, who gave the following answer [Topology, 9 (1970), 283-298].

Theorem 1 (Murasugi).

Recall [4] that the invariant ξ(l) is defined by first orienting l, giving, an oriented link l, say, and then setting ξ(l) = σ(l) + Lk(k₁,k₂), where σ denotes signature and Lk linking number.

In the present note we shall give an alternative, more conceptual, proof of Theorem 1, and in fact obtain it as a special case of a considerably more general result.

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 17 April 1978

Published: 1 May 1979