Vol. 90, No. 1, 1980

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

Jonathan Simon

Vol. 90 (1980), No. 1, 177–190

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
Received: 3 October 1978
Revised: 21 June 1979
Published: 1 September 1980
Jonathan Simon