Totally knotted Seifert surfaces with accidental peripherals
HTML articles powered by AMS MathViewer
- by Makoto Ozawa and Yukihiro Tsutsumi PDF
- Proc. Amer. Math. Soc. 131 (2003), 3945-3954 Request permission
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
- M. Culler, C. McA. Gordon, J. Luecke, and P. B. Shalen, Correction to: “Dehn surgery on knots” [Ann. of Math. (2) 125 (1987), no. 2, 237–300; MR0881270 (88a:57026)], Ann. of Math. (2) 127 (1988), no. 3, 663. MR 942524, DOI 10.2307/2007009
- Sérgio R. Fenley, Quasi-Fuchsian Seifert surfaces, Math. Z. 228 (1998), no. 2, 221–227. MR 1630563, DOI 10.1007/PL00004607
- David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
- Richard F. Gustafson, A simple genus one knot with incompressible spanning surfaces of arbitrarily high genus, Pacific J. Math. 96 (1981), no. 1, 81–98. MR 634764, DOI 10.2140/pjm.1981.96.81
- Kazuhiro Ichihara and Makoto Ozawa, Accidental surfaces in knot complements, J. Knot Theory Ramifications 9 (2000), no. 6, 725–733. MR 1775383, DOI 10.1142/S0218216500000414
- William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450, DOI 10.1090/cbms/043
- Masaharu Kouno, Kimihiko Motegi, and Tetsuo Shibuya, Twisting and knot types, J. Math. Soc. Japan 44 (1992), no. 2, 199–216. MR 1154840, DOI 10.2969/jmsj/04420199
- Herbert C. Lyon, Incompressible surfaces in knot spaces, Trans. Amer. Math. Soc. 157 (1971), 53–62. MR 275412, DOI 10.1090/S0002-9947-1971-0275412-6
- W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1978.
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
- Email: ozawa@musubime.com
- Yukihiro Tsutsumi
- Affiliation: Department of Mathematics, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan
- Email: yukihiro@math.keio.ac.jp
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3945-3954
- MSC (2000): Primary 57M25; Secondary 57N10
- DOI: https://doi.org/10.1090/S0002-9939-03-06964-8
- MathSciNet review: 1999945