Wirtinger approximations and the knot groups of Fⁿ in Sⁿ⁺²

Jonathan Simon

Vol. 90 (1980), No. 1, 177–190

DOI: 10.2140/pjm.1980.90.177

Abstract

We consider the problem of deciding whether or not a given group G has a Wirtinger presentation, i.e., a presentation in which each defining relation states that two generators are conjugate or that a generator commutes with some word. This property is important because it characterizes those groups that can be realized as knot groups of closed, orientable n-manifolds in Sⁿ⁺². We isolate the obstruction in the form of an abelian group somewhat related to H₂(G). We do this by considering Wirtinger-presented groups that are approximations to G and prove the existence of a best-approximation.

Mathematical Subject Classification 2000

Primary: 57M05

Secondary: 57Q45

Milestones

Received: 3 October 1978

Revised: 21 June 1979

Published: 1 September 1980