Thin position and bridge number for knots in the 3-sphere. (English) Zbl 0867.57009

Thompson, Abigail

Topology 36, No. 2, 505-507 (1997).

The bridge number of a knot in \(S^3\), introduced by H. Schubert [Math. Z. 61, 245-288 (1954; Zbl 0058.17403)] is a classical and well-understood knot invariant. The concept of thin position for a knot was developed fairly recently by D. Gabai [J. Differ. Geom. 26, 479-536 (1987; Zbl 0639.57008)]. It has proved to be an extraordinarily useful notion, playing a key role in Gabai’s proof of property \(R\) as well as Gordon-Luecke’s solution of the knot complement problem [C. McA. Gordon and J. Luecke, J. Am. Math. Soc. 2, 371-415 (1989; Zbl 0678.57005)]. The purpose of this paper is to examine the relation between bridge number and thin position. We show that either a knot in thin position is also in the position which realizes its bridge number or the knot has an incompressible meridianal planar surface properly imbedded in its complement. The second possibility implies that the knot has a generalized tangle decomposition along an incompressible punctured 2-sphere. Using results from [M. Culler, C. McA. Gordon, J. Luecke and P. B. Shalen, Ann. Math., II. Ser. 125, 237-300 (1987; Zbl 0633.57006); C. McA. Gordon and A. Reid, J. Knot Theory Ramifications 4, 389-409 (1995; Zbl 0841.57012)], we note that if thin position for a knot \(K\) is not bridge position, then there exists a closed incompressible surface in the complement of the knot (Corollary 3) and that the tunnel number of the knot is strictly greater than one (Corollary 4).

MSC:

57M25

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

Keywords:

bridge number; knot; thin position; knot complement problem; tangle decomposition; tunnel number

Citations:

Zbl 0678.57005; Zbl 0639.57008; Zbl 0058.17403; Zbl 0633.57006; Zbl 0841.57012