International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 2, Pages 289-292

Geometric presentations of classical knot groups

John Erbland1 and Mauricio Guterriez2

1Department of Mathematics, University of Hartford, West Hartford 06117, Connecticut, USA
2Mathematics Department, Tufts University, Medford 02155, Massachusetts, USA

Received 19 September 1989

Copyright © 1991 John Erbland and Mauricio Guterriez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The question addressed by thls paper is, how close is the tunnel number of a knot to the minimum number of relators in a presentation of the knot group? A dubious, but useful conjecture, is that these two invariants are equal. (The analogous assertion applied to 3-manifolds is known to be false. [1]). It has been shown recently [2] that not all presentations of a knot group are “geometric”. The main result in this paper asserts that the tunnel number is equal to the minimum number of relators among presentations satisfying a somewhat restrictive condition, that is, that such presentations are always geometric.