Dehn surgery on knots of wrapping number 2

Wu, Ying-Qing

doi:10.2140/agt.2013.13.479

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Dehn surgery on knots of wrapping number $2$

Ying-Qing Wu

Algebraic & Geometric Topology 13 (2013) 479–503

DOI: 10.2140/agt.2013.13.479

Abstract

Suppose $K$ is a hyperbolic knot in a solid torus $V$ intersecting a meridian disk $D$ twice. We will show that if $K$ is not the Whitehead knot and the frontier of a regular neighborhood of $K \cup D$ is incompressible in the knot exterior, then $K$ admits at most one exceptional surgery, which must be toroidal. Embedding $V$ in $S^{3}$ gives infinitely many knots $K_{n}$ with a slope $r_{n}$ corresponding to a slope $r$ of $K$ in $V$ . If $r$ surgery on $K$ in $V$ is toroidal then either $K_{n} (r_{n})$ are toroidal for all but at most three $n$ , or they are all atoroidal and nonhyperbolic. These will be used to classify exceptional surgeries on wrapped Montesinos knots in a solid torus, obtained by connecting the top endpoints of a Montesinos tangle to the bottom endpoints by two arcs wrapping around the solid torus.

Keywords

Exceptional Dhen Surgery, hyperbolic manifolds, wrapping number

Mathematical Subject Classification 2010

Primary: 57N10

References

Publication

Received: 25 September 2011

Revised: 24 July 2012

Accepted: 4 October 2012

Published: 6 March 2013

Authors

Ying-Qing Wu
Department of Mathematics The University of Iowa 14 MacLean Hall Iowa City, IA 52242-1419 USA

Volume 13, issue 1 (2013)