Dehn fillings of knot manifolds containing essential once-punctured tori
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- by Steven Boyer, Cameron McA. Gordon and Xingru Zhang PDF
- Trans. Amer. Math. Soc. 366 (2014), 341-393 Request permission
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 $|n|>1$. 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.References
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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
- MR Author ID: 219677
- Email: boyer.steven@uqam.ca
- Cameron McA. Gordon
- Affiliation: Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, Texas 78712
- MR Author ID: 75435
- Email: gordon@math.utexas.edu
- Xingru Zhang
- Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14214-3093
- MR Author ID: 346629
- Email: xinzhang@buffalo.edu
- 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. - © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 341-393
- MSC (2010): Primary 57M25, 57M50, 57M99
- DOI: https://doi.org/10.1090/S0002-9947-2013-05837-0
- MathSciNet review: 3118399