Stabilizing Heegaard splittings of high-distance knots

Mossessian, George

doi:10.2140/agt.2016.16.3419

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Stabilizing Heegaard splittings of high-distance knots

George Mossessian

Algebraic & Geometric Topology 16 (2016) 3419–3443

DOI: 10.2140/agt.2016.16.3419

Abstract

Suppose $K$ is a knot in $S^{3}$ with bridge number $n$ and bridge distance greater than $2 n$ . We show that there are at most $(\binom{2 n}{n})$ distinct minimal-genus Heegaard splittings of $S^{3} ∖ η (K)$ . These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4 n$ , then two splittings from different families become equivalent only after $n - 1$ stabilizations. Furthermore, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to these Heegaard surfaces.

Keywords

knot complement, high distance, common stabilization, Heegaard splitting, tunnel system

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27

References

Publication

Received: 8 September 2015

Revised: 15 February 2016

Accepted: 21 April 2016

Published: 15 December 2016

Authors

George Mossessian
Department of Mathematics University of California, Davis 1 Shields Avenue Davis, CA 95616 United States
http://math.ucdavis.edu/~gmoss

Volume 16, issue 6 (2016)