Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Essential surfaces of non-negative Euler characteristic in genus two handlebody exteriors


Authors: Yuya Koda and Makoto Ozawa; with an appendix by Cameron Gordon
Journal: Trans. Amer. Math. Soc. 367 (2015), 2875-2904
MSC (2010): Primary 57M25; Secondary 57M15
DOI: https://doi.org/10.1090/S0002-9947-2014-06199-0
Published electronically: October 3, 2014
MathSciNet review: 3301885
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide a classification of the essential surfaces of non-negative Euler characteristic in the exteriors of genus two handlebodies embedded in the 3-sphere.


References [Enhancements On Off] (What's this?)

  • [1] Erol Akbas, A presentation for the automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Pacific J. Math. 236 (2008), no. 2, 201-222. MR 2407105 (2009d:57029), https://doi.org/10.2140/pjm.2008.236.201
  • [2] R. Blair, M. Campisi, J. Johnson, S. A. Taylor, and M. Tomova, Genus bounds bridge number for high distance knots, arXiv:1211.4787.
  • [3] R. Benedetti and R. Frigerio, Levels of knotting of spatial handlebodies, Trans. Amer. Math. Soc. 365 (2013), no. 4, 2099-2167. MR 3009654, https://doi.org/10.1090/S0002-9947-2012-05707-2
  • [4] Francis Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 237-270. MR 732345 (85j:57011)
  • [5] Gerhard Burde and Heiner Zieschang, Knots, de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776 (87b:57004)
  • [6] Sangbum Cho, Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1113-1123 (electronic). MR 2361888 (2009c:57029), https://doi.org/10.1090/S0002-9939-07-09188-5
  • [7] Mario Eudave Muñoz, Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc. 330 (1992), no. 2, 463-501. MR 1112545 (92m:57009), https://doi.org/10.2307/2153918
  • [8] Mario Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35-61. MR 1470720 (98i:57007)
  • [9] Mario Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 119-141. MR 1903687 (2003c:57005), https://doi.org/10.1016/S0166-8641(01)00114-6
  • [10] M. Eudave-Muñoz and M. Ozawa, Composite tunnel number one genus two handlebody-knots, Bol. Soc. Mat. Mexicana (2014), DOI 10.1007/s40590-014-0035-5.
  • [11] David Gabai, Foliations and the topology of $ 3$-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479-536. MR 910018 (89a:57014b)
  • [12] Lebrecht Goeritz, Die abbildungen der brezelfläche und der vollbrezel vom geschlecht 2, Abh. Math. Sem. Univ. Hamburg 9 (1933), no. 1, 244-259 (German). MR 3069602, https://doi.org/10.1007/BF02940650
  • [13] Francisco González-Acuña and Hamish Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 89-102. MR 809502 (87c:57003), https://doi.org/10.1017/S0305004100063969
  • [14] C. McA. Gordon, On primitive sets of loops in the boundary of a handlebody, Topology Appl. 27 (1987), no. 3, 285-299. MR 918538 (88k:57013), https://doi.org/10.1016/0166-8641(87)90093-9
  • [15] C. McA. Gordon and R. A. Litherland, Incompressible planar surfaces in $ 3$-manifolds, Topology Appl. 18 (1984), no. 2-3, 121-144. MR 769286 (86e:57013), https://doi.org/10.1016/0166-8641(84)90005-1
  • [16] C. McA. Gordon and J. Luecke, Only integral Dehn surgeries can yield reducible manifolds, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 97-101. MR 886439 (89a:57003), https://doi.org/10.1017/S0305004100067086
  • [17] C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371-415. MR 965210 (90a:57006a), https://doi.org/10.2307/1990979
  • [18] C. McA. Gordon and John Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004), no. 1-2, 417-485. MR 2074884 (2005k:57013)
  • [19] Wolfgang Haken, Some results on surfaces in $ 3$-manifolds, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39-98. MR 0224071 (36 #7118)
  • [20] Chuichiro Hayashi and Koya Shimokawa, Symmetric knots satisfy the cabling conjecture, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 3, 501-529. MR 1607989 (99c:57022), https://doi.org/10.1017/S0305004197002399
  • [21] John Hempel, $ 3$-Manifolds, Princeton University Press, Princeton, N. J., 1976. Ann. of Math. Studies, No. 86. MR 0415619 (54 #3702)
  • [22] James Allen Hoffman, Reducing spheres and embedded projective planes after Dehn surgery on a knot, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)-The University of Texas at Austin. MR 2693370
  • [23] James A. Hoffman, There are no strict great $ x$-cycles after a reducing or $ P^2$ surgery on a knot, J. Knot Theory Ramifications 7 (1998), no. 5, 549-569. MR 1637581 (99h:57008), https://doi.org/10.1142/S0218216598000309
  • [24] Kazuhiro Ichihara and Makoto Ozawa, Accidental surfaces in knot complements, J. Knot Theory Ramifications 9 (2000), no. 6, 725-733. MR 1775383 (2001f:57007), https://doi.org/10.1142/S0218216500000414
  • [25] Atsushi Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol. 8 (2008), no. 3, 1403-1418. MR 2443248 (2009g:57022), https://doi.org/10.2140/agt.2008.8.1403
  • [26] A. Ishii, K. Kishimoto, and M. Ozawa, Knotted handle decomposing spheres for handlebody-knots, to appear in J. Math. Soc. Japan.
  • [27] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450 (81k:57009)
  • [28] Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston Inc., Boston, MA, 2001. MR 1792613 (2002m:57018)
  • [29] H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresber. Deutsch. Math.-Verein. 38 (1929), 248-260.
  • [30] Tsuyoshi Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21 (1984), no. 2, 437-455. MR 752472 (85k:57011)
  • [31] Y. Koda, Automorphisms of the $ 3$-sphere that preserve spatial graphs and handlebody-knots, arXiv:1106.4777.
  • [32] Jung Hoon Lee and Sangyop Lee, Inequivalent handlebody-knots with homeomorphic complements, Algebr. Geom. Topol. 12 (2012), no. 2, 1059-1079. MR 2928904, https://doi.org/10.2140/agt.2012.12.1059
  • [33] E. Luft and X. Zhang, Symmetric knots and the cabling conjecture, Math. Ann. 298 (1994), no. 3, 489-496. MR 1262772 (95f:57014), https://doi.org/10.1007/BF01459747
  • [34] William W. Menasco and Morwen B. Thistlethwaite, Surfaces with boundary in alternating knot exteriors, J. Reine Angew. Math. 426 (1992), 47-65. MR 1155746 (93d:57019)
  • [35] John W. Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 37-125. MR 758464, https://doi.org/10.1016/S0079-8169(08)61637-2
  • [36] Michael Motto, Inequivalent genus $ 2$ handlebodies in $ S^3$ with homeomorphic complement, Topology Appl. 36 (1990), no. 3, 283-290. MR 1070707 (91j:57008), https://doi.org/10.1016/0166-8641(90)90052-4
  • [37] J-P. Otal, Le théorème d'hyperbolisation pour les variétés fibrées de dimension $ 3$, Astérisque 235, 1996. MR 1402300 (97e:57013)
  • [38] Jean-Pierre Otal, Thurston's hyperbolization of Haken manifolds, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996) Int. Press, Boston, MA, 1998, pp. 77-194. MR 1677888 (2000b:57025)
  • [39] Walter Parry, All types implies torsion, Proc. Amer. Math. Soc. 110 (1990), no. 4, 871-875. MR 1039537 (91c:20079), https://doi.org/10.2307/2047731
  • [40] Józef H. Przytycki, Incompressibility of surfaces after Dehn surgery, Michigan Math. J. 30 (1983), no. 3, 289-308. MR 725782 (86g:57012), https://doi.org/10.1307/mmj/1029002906
  • [41] Martin Scharlemann, Producing reducible $ 3$-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481-500. MR 1071370 (91i:57003), https://doi.org/10.1016/0040-9383(90)90017-E
  • [42] Martin Scharlemann, Automorphisms of the $ 3$-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. Special Issue, 503-514. MR 2199366 (2007c:57020)
  • [43] Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), no. 3, 57-104 (German). MR 0031733 (11,196f)
  • [44] William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR 648524 (83h:57019), https://doi.org/10.1090/S0273-0979-1982-15003-0
  • [45] Yasuyuki Tsukui, On a prime surface of genus $ 2$ and homeomorphic splitting of $ 3$-sphere, Yokohama Math J. 23 (1975), no. 1-2, 63-75. MR 0377888 (51 #14057)
  • [46] Friedhelm Waldhausen, Heegaard-Zerlegungen der $ 3$-sphäre, Topology 7 (1968), 195-203 (German). MR 0227992 (37 #3576)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57M25, 57M15

Retrieve articles in all journals with MSC (2010): 57M25, 57M15


Additional Information

Yuya Koda
Affiliation: Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan
Email: koda@math.tohoku.ac.jp

Makoto Ozawa
Affiliation: Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan
Email: w3c@komazawa-u.ac.jp

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

DOI: https://doi.org/10.1090/S0002-9947-2014-06199-0
Keywords: Knot, handlebody, essential surface
Received by editor(s): February 21, 2013
Received by editor(s) in revised form: May 29, 2013
Published electronically: October 3, 2014
Additional Notes: The first-named author was supported in part by Grant-in-Aid for Young Scientists (B) (No. 26800028), Japan Society for the Promotion of Science, and by JSPS Postdoctoral Fellowships for Research Abroad
The second-named author was supported in part by Grant-in-Aid for Scientific Research (C) (No. 23540105), Japan Society for the Promotion of Science.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society