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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dehn fillings of knot manifolds containing essential once-punctured tori


Authors: Steven Boyer, Cameron McA. Gordon and Xingru Zhang
Journal: Trans. Amer. Math. Soc. 366 (2014), 341-393
MSC (2010): Primary 57M25, 57M50, 57M99
Published electronically: June 10, 2013
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Abstract: In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $ M$ be such a knot manifold and let $ \beta $ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $ M$ with slope $ \alpha $ produces a Seifert fibred manifold, then $ \Delta (\alpha ,\beta )\leq 5$. Furthermore we classify the triples $ (M; \alpha ,\beta )$ when $ \Delta (\alpha ,\beta )\geq 4$. More precisely, when $ \Delta (\alpha ,\beta )=5$, then $ M$ is the (unique) manifold $ Wh(-3/2)$ obtained by Dehn filling one boundary component of the Whitehead link exterior with slope $ -3/2$, and $ (\alpha , \beta )$ is the pair of slopes $ (-5, 0)$. Further, $ \Delta (\alpha ,\beta )=4$ if and only if $ (M; \alpha ,\beta )$ is the triple $ \displaystyle (Wh(\frac {-2n\pm 1}{n}); -4, 0)$ for some integer $ n$ with $ \vert n\vert>1$. Combining this with known results, we classify all hyperbolic knot manifolds $ M$ and pairs of slopes $ (\beta , \gamma )$ on $ \partial M$ where $ \beta $ is the boundary slope of an essential once-punctured torus in $ M$ and $ \gamma $ is an exceptional filling slope of distance $ 4$ or more from $ \beta $. Refined results in the special case of hyperbolic genus one knot exteriors in $ S^3$ are also given.


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Additional Information

Steven Boyer
Affiliation: Département de Mathématiques, Université du Québec à Montréal, 201 avenue du Président-Kennedy, Montréal, Québec, Canada H2X 3Y7
Email: boyer.steven@uqam.ca

Cameron McA. Gordon
Affiliation: Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, Texas 78712
Email: gordon@math.utexas.edu

Xingru Zhang
Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14214-3093
Email: xinzhang@buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05837-0
PII: S 0002-9947(2013)05837-0
Received by editor(s): November 8, 2011
Received by editor(s) in revised form: March 23, 2012
Published electronically: June 10, 2013
Additional Notes: The first author was partially supported by NSERC grant RGPIN 9446-2008
The second author was partially supported by NSF grant DMS-0906276.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.