Multiple bridge surfaces restrict knot distance

Details Multiple bridge surfaces restrict knot distance Multiple bridge surfaces restrict knot distance

2005-11-05

arxiv.org/abs/math/0511139v3

Mathematics

Geometric Topology

47 pages, 16 figures, references and figures added, the main result has been slightly strengthened

Scientific paper

Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M, P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or $d(K,P) \leq 2-\chi(Q-K)$. If K is not a two bridge knot, then the result holds even if K is removable with respect to Q. As a corollary we show that if a knot in the 3-sphere has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position.

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