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Geometric presentations of classical knot groups. (English) Zbl 0728.57003
Summary: The question addressed by this 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 [J. M. Montesinos, Low dimensional topology and Kleinian groups, Symp. Warwick and Durham 1984, Lond. Math. Soc. Lect. Note Ser. 112, 241-252 (1986; Zbl 0619.57003)].) It has been shown recently [M. Boileau, M. Rost, and H. Zieschang, Math. Ann. 279, No.3, 553-581 (1988; Zbl 0616.57008)] 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.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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