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

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: -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.

**[BZ]**Gerhard Burde and Heiner Zieschang,*Knots*, de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR**808776****[EM]**Mario Eudave Muñoz,*On nonsimple 3-manifolds and 2-handle addition*, Topology Appl.**55**(1994), no. 2, 131–152. MR**1256216**, 10.1016/0166-8641(94)90114-7**[EU]**Jun Murakami,*Kontsevich’s integral for the Homfly polynomial and its applications*, Sūrikaisekikenkyūsho Kōkyūroku**883**(1994), 134–147. Geometric aspects of infinite integrable systems (Japanese) (Kyoto, 1993). MR**1338872****[G]**David Gabai,*Foliations and the topology of 3-manifolds. III*, J. Differential Geom.**26**(1987), no. 3, 479–536. MR**910018****[GST]**Hiroshi Goda, Martin Scharlemann, and Abigail Thompson,*Levelling an unknotting tunnel*, Geom. Topol.**4**(2000), 243–275 (electronic). MR**1778174**, 10.2140/gt.2000.4.243**[GT]**Hiroshi Goda and Masakazu Teragaito,*Tunnel number one genus one non-simple knots*, Tokyo J. Math.**22**(1999), no. 1, 99–103. MR**1692023**, 10.3836/tjm/1270041615**[GL]**C. McA. Gordon and R. A. Litherland,*Incompressible planar surfaces in 3-manifolds*, Topology Appl.**18**(1984), no. 2-3, 121–144. MR**769286**, 10.1016/0166-8641(84)90005-1**[Ma]**H. Matsuda,*Genus one knots which admit -decompositions*, Proc. AMS.**130**(2001), 2155-2163.**[Mo]**Kanji Morimoto,*Planar surfaces in a handlebody and a theorem of Gordon-Reid*, KNOTS ’96 (Tokyo), World Sci. Publ., River Edge, NJ, 1997, pp. 123–146. MR**1664957****[MS]**Kanji Morimoto and Makoto Sakuma,*On unknotting tunnels for knots*, Math. Ann.**289**(1991), no. 1, 143–167. MR**1087243**, 10.1007/BF01446565**[OZ]**R. P. Osborne and H. Zieschang,*Primitives in the free group on two generators*, Invent. Math.**63**(1981), no. 1, 17–24. MR**608526**, 10.1007/BF01389191**[Sc]**Martin Scharlemann,*Outermost forks and a theorem of Jaco*, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 189–193. MR**813113**, 10.1090/conm/044/813113**[Sc2]**M. Scharlemann,*The Goda-Teragaito Conjecture: an overview*, Surikaisekikenkyusho Kokyuroku**1229**(2001) 87-102 or http://front.math.ucdavis.edu/math.GT/0108079.**[ST1]**M. Scharlemann and A. Thompson,*Unknotting tunnels and Seifert surfaces*, preprint (http://front.math.ucdavis.edu/math.GT/0010212).**[ST2]**M. Scharlemann and A. Thompson,*Thinning genus two Heegaard spines in*, to appear.**[ST3]**Martin Scharlemann and Abigail Thompson,*Thin position and Heegaard splittings of the 3-sphere*, J. Differential Geom.**39**(1994), no. 2, 343–357. MR**1267894****[T]**Abigail Thompson,*Thin position and bridge number for knots in the 3-sphere*, Topology**36**(1997), no. 2, 505–507. MR**1415602**, 10.1016/0040-9383(96)00010-9

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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