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Assume $V$ is a handlebody and $C$ be a simple closed curve contained in the interior of $V$. As Sam said, there exists some simple closed curve such that every dehn surgery along it produces a handlebody. So I assume $C$ can not be isotopic to a simple closed curve in $\partial V$.

Obviously, trivial dehn surgery along $C$ produces a handlebody. So My question is:

Is there a different dehn surgery along $C$ which produces a handlebody?

Can we classify all the dehn surgery along $C$ which produce handlebodies?

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With respect, the third line is unclear and needs to be rewritten. Also, it would help to have a definition of unknotted. – Sam Nead Dec 28 '11 at 10:58
I can point out that if $C$ is a core curve (meets some essential disk in a single point and is also isotopic into the boundary) then every Dehn filling on $C$ gives a handlebody. – Sam Nead Dec 28 '11 at 11:01
@Sam. Thank you for your advice. I have rewritten it. – yanqing Dec 28 '11 at 11:44
@Yanqing - There are many pairs $(V, C)$ where we may embed $V$ into three-space and $V$ is a standard handlebody, $C$ is an unknot in three-space, and yet $C$ is very complicated inside of $V$. So, with your current definition, I see no hope for a reasonable answer to your question. That said, you might be interested in the ideas of "cosmetic Dehn surgery" and "Property R". best – Sam Nead Dec 28 '11 at 12:29
@Sam. Lots of thanks. – yanqing Dec 28 '11 at 13:31
up vote 13 down vote accepted

There's an extensive literature on this and more general questions.

First, let's consider a more precise formulation of the question. Let $H$ be a handlebody, and let $K\subset H$ be a knot. Let's assume that $\partial H \subset H-K$ is incompressible. Otherwise, $H=H_0 \natural H_1$ the boundary connect sum of two handlebodies, such that $K\subset H_0$, and $\partial H_0 \subset H_0-K$ is incompressible. Then a surgery on $K$ makes $H$ a handlebody if and only if the corresponding surgery makes $H_0$ a handlebody. Thus, one may reduce to considering the case that $K$ is diskbusting.

With this reformulation of the question, the case that $K$ is a core curve corresponds to when $H$ is a solid torus, and $K$ is a core curve (since this is the only case in which a core curve is diskbusting). So assume that $K$ is not a core curve. If $K$ is isotopic into $\partial H$, then there are infinitely many surgeries which yield $H$. In this case, there is an annulus going between the knot and $\partial H$. The surgery slopes which intersect the annulus slope once give back $H$. Surgery along the annular slope gives a manifold containing an incompressible surface, by Jaco's lemma. If the intersection with the annulus slope is $>1$, then the manifold has incompressible boundary by Theorem 2.4.3 of the cyclic surgery paper.

A result of Wu implies that for knots $K$ which are not isotopic into the boundary, the distance between boundary-reducible surgeries is at most one, and therefore there are at most two non-trivial surgeries which may yield a handlebody.

In the case that $H$ is a solid torus, it was proved by Berge and Gabai that $K$ must be a 1-bridge braid, and a complete description was given. There is a famous example (the Berge link) which yields 3 solid torus surgeries, which shows that Wu's estimate is sharp (there is a 3-fold symmetry permuting the slopes).

Wu has further results on the case of 1-bridge knots in handlebodies. Added: Frigerio, Martelli and Petronio show that there are 1-bridge knots in handlebodies with three handlebody fillings (including the trivial one), generalizing the example of the Berge link to higher genus boundary, and showing that Wu's theorem is sharp in general. It seems to be an open question whether all such examples are 1-bridge (see discussion in another of Wu's papers).

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In a joint paper with Martelli and Petronio (Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary, J. Diff. Geom. 64 (2003), 425-456), for every g>1 we gave examples of knots in the genus-g handlebody with the following properties: the complement of each such knot is hyperbolic (with one cusp and a geodesic boundary component), so in particular each such knot is not isotopic to the boundary of the handlebody; each of them has exactly three surgeries giving back the handlebody. Every knot in our family is 1-bridge. – Roberto Frigerio Dec 28 '11 at 22:58
Thanks for the reference Roberto, sorry for overlooking your result! – Ian Agol Dec 29 '11 at 0:17
@Agol. In the last second paragraph, $H$ should be a handlebody ? – yanqing Dec 29 '11 at 2:56
With this reformulation of the question, the case that K is a core curve corresponds to when H is a solid torus, and K is a core curve. Are two $K$ same above? – yanqing Dec 29 '11 at 3:01
@Agol: R. Sean Bowman constructed knots in a genus 2 handle body that have a non-trivial handle body filling but are not 1-bridge. His results are here: – Neil Hoffman Dec 19 '12 at 13:40

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