Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Totally knotted Seifert surfaces with accidental peripherals

Authors: Makoto Ozawa and Yukihiro Tsutsumi
Journal: Proc. Amer. Math. Soc. 131 (2003), 3945-3954
MSC (2000): Primary 57M25; Secondary 57N10
Published electronically: April 30, 2003
MathSciNet review: 1999945
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that if there exists an essential accidental surface in the knot exterior, then a closed accidental surface also exists. As its corollary, we know boundary slopes of accidental essential surfaces are integral or meridional. It is shown that an accidental incompressible Seifert surface in knot exteriors in $S^3$ is totally knotted. Examples of satellite knots with arbitrarily high genus Seifert surfaces with accidental peripherals are given, and a Haken 3-manifold which contains a hyperbolic knot with an accidental incompressible Seifert surface of genus one is also given.

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

  • 1. M. Culler, C. Gordon, J. Luecke, and P. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300. MR 89c:57015
  • 2. S. R. Fenley, Quasi-Fuchsian Seifert surfaces, Math. Z. 228 (1998), 221-227. MR 99c:57037
  • 3. D. Gabai, Foliations and the topology of 3-manifolds III, J. Diff. Geom. 26 (1987), 479-536. MR 89a:57014b
  • 4. R. F. Gustafson, A simple genus one knot with incompressible spanning surfaces of arbitrarily high genus, Pacific J. Math. 96 (1981), 81-98. MR 83a:57008
  • 5. K. Ichihara and M. Ozawa, Accidental surfaces in knot complements, J. Knot Theory and its Ramifications, 9 (2000), 725-733. MR 2001f:57007
  • 6. W. Jaco, Lectures on Three Manifold Topology, AMS Conference board of Math. No. 43, 1980. MR 81k:57009
  • 7. M. Kouno, K. Motegi and T. Shibuya, Twisting and knot types, J. Math. Soc. Japan 44 (1992), 199-216. MR 93e:57011
  • 8. H. C. Lyon, Incompressible surfaces in knot spaces, Trans. Amer. math. soc, 157 (1971), 53-62. MR 43:1169
  • 9. W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1978.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M25, 57N10

Retrieve articles in all journals with MSC (2000): 57M25, 57N10

Additional Information

Makoto Ozawa
Affiliation: Department of Mathematics, School of Education, Waseda University, Nishiwaseda 1-6-1, Shinjuku-ku, Tokyo 169-8050, Japan
Address at time of publication: Natural Science Faculty, Faculty of Letters, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan

Yukihiro Tsutsumi
Affiliation: Department of Mathematics, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan

Keywords: Accidental peripheral, quasi-Fuchsian, Seifert surface, totally knotted
Received by editor(s): July 25, 2000
Received by editor(s) in revised form: July 25, 2002
Published electronically: April 30, 2003
Additional Notes: The first author was supported in part by Fellowship of the Japan Society for the Promotion of Science for Japanese Junior Scientists
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society