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There are no unexpected tunnel number one knots of genus one


Author: Martin Scharlemann
Journal: Trans. Amer. Math. Soc. 356 (2004), 1385-1442
MSC (2000): Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-03-03182-9
Published electronically: October 6, 2003
MathSciNet review: 2034312
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Abstract: We show that the only knots that are tunnel number one and genus one are those that are already known: $2$-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Muñoz and by Morimoto and Sakuma. This confirms a conjecture first made by Goda and Teragaito.


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Additional Information

Martin Scharlemann
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: mgscharl@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03182-9
Received by editor(s): July 24, 2001
Received by editor(s) in revised form: July 25, 2002
Published electronically: October 6, 2003
Additional Notes: This research was supported in part by an NSF grant, the Miller Institute, and RIMS Kyoto
Article copyright: © Copyright 2003 American Mathematical Society

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