Geometric presentations of classical knot groups - John Erbland; Mauricio Guterriez (International Journal of Mathematics and Mathematical Sciences)
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.