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

DOI:
https://doi.org/10.1090/S0002-9947-2013-05837-0

Published electronically:
June 10, 2013

MathSciNet review:
3118399

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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 be such a knot manifold and let be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling with slope produces a Seifert fibred manifold, then . Furthermore we classify the triples when . More precisely, when , then is the (unique) manifold obtained by Dehn filling one boundary component of the Whitehead link exterior with slope , and is the pair of slopes . Further, if and only if is the triple for some integer with . Combining this with known results, we classify all hyperbolic knot manifolds and pairs of slopes on where is the boundary slope of an essential once-punctured torus in and is an exceptional filling slope of distance or more from . Refined results in the special case of hyperbolic genus one knot exteriors in 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:
https://doi.org/10.1090/S0002-9947-2013-05837-0

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.