Obtaining genus 2 Heegaard splittings from Dehn surgery

Baker, Kenneth L; Gordon, Cameron; Luecke, John

doi:10.2140/agt.2013.13.2471

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Obtaining genus $2$ Heegaard splittings from Dehn surgery

Kenneth L Baker, Cameron Gordon and John Luecke

Algebraic & Geometric Topology 13 (2013) 2471–2634

DOI: 10.2140/agt.2013.13.2471

Abstract

Let $K^{'}$ be a hyperbolic knot in $S^{3}$ and suppose that some Dehn surgery on $K^{'}$ with distance at least $3$ from the meridian yields a $3$ –manifold $M$ of Heegaard genus $2$ . We show that if $M$ does not contain an embedded Dyck’s surface (the closed nonorientable surface of Euler characteristic $- 1$ ), then the knot dual to the surgery is either $0$ –bridge or $1$ –bridge with respect to a genus $2$ Heegaard splitting of $M$ . In the case that $M$ does contain an embedded Dyck’s surface, we obtain similar results. As a corollary, if $M$ does not contain an incompressible genus $2$ surface, then the tunnel number of $K^{'}$ is at most $2$ .

Keywords

Dehn surgery, bridge number, Heegaard splitting

Mathematical Subject Classification 2010

Primary: 57M27

References

Publication

Received: 1 May 2012

Revised: 6 February 2013

Accepted: 7 March 2013

Published: 5 July 2013

Authors

Kenneth L Baker
Department of Mathematics University of Miami PO Box 249085 Coral Gables, FL 33146 USA
http://math.miami.edu/~kenken
Cameron Gordon
Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin, TX 78712-1202 USA
http://www.ma.utexas.edu/text/webpages/gordon.html
John Luecke
Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257 USA

Volume 13, issue 5 (2013)