Volume 7, issue 2 (2007)

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Multiple bridge surfaces restrict knot distance

Maggy Tomova

Algebraic & Geometric Topology 7 (2007) 957–1006
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) 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 S3 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