Multiple bridge surfaces restrict knot distance

Tomova, Maggy

doi:10.2140/agt.2007.7.957

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Multiple bridge surfaces restrict knot distance

Maggy Tomova

Algebraic & Geometric Topology 7 (2007) 957–1006

DOI: 10.2140/agt.2007.7.957

Abstract

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 $d (K, P) \leq 2 - χ (Q - K)$ . If $K$ is not a 2–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 $S^{3}$ has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position.

Keywords

knot distance, bridge position, Heegaard splitting, strongly irreducible, weakly incompressible

Mathematical Subject Classification 2000

Primary: 57M25, 57M27, 57M50

References

Publication

Received: 5 April 2007

Accepted: 7 May 2007

Published: 20 June 2007

Authors

Maggy Tomova
Mathematics Department Rice University 6100 S Main Street Houston TX 77005-1892 USA

Volume 7, issue 2 (2007)