#### Volume 13, issue 1 (2013)

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Dehn surgery on knots of wrapping number $2$

### Ying-Qing Wu

Algebraic & Geometric Topology 13 (2013) 479–503
##### 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 ${K}_{n}\left({r}_{n}\right)$ 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
Primary: 57N10
##### 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