Totally knotted Seifert surfaces with accidental peripherals
Makoto Ozawa and Yukihiro Tsutsumi
Proc. Amer. Math. Soc. 131 (2003), 3945-3954
Primary 57M25; Secondary 57N10
April 30, 2003
Full-text PDF Free Access
Similar Articles |
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 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.
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
R. Fenley, Quasi-Fuchsian Seifert surfaces, Math. Z.
228 (1998), no. 2, 221–227. MR 1630563
Gabai, Foliations and the topology of 3-manifolds. III, J.
Differential Geom. 26 (1987), no. 3, 479–536.
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
Ichihara and Makoto
Ozawa, Accidental surfaces in knot complements, J. Knot Theory
Ramifications 9 (2000), no. 6, 725–733. MR 1775383
Jaco, Lectures on three-manifold topology, CBMS Regional
Conference Series in Mathematics, vol. 43, American Mathematical
Society, Providence, R.I., 1980. MR 565450
Motegi, and Tetsuo
Shibuya, Twisting and knot types, J. Math. Soc. Japan
44 (1992), no. 2, 199–216. MR 1154840
C. Lyon, Incompressible surfaces in knot
spaces, Trans. Amer. Math. Soc. 157 (1971), 53–62. MR 0275412
(43 #1169), http://dx.doi.org/10.1090/S0002-9947-1971-0275412-6
W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1978.
- M. Culler, C. Gordon, J. Luecke, and P. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300. MR 89c:57015
- S. R. Fenley, Quasi-Fuchsian Seifert surfaces, Math. Z. 228 (1998), 221-227. MR 99c:57037
- D. Gabai, Foliations and the topology of 3-manifolds III, J. Diff. Geom. 26 (1987), 479-536. MR 89a:57014b
- 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
- K. Ichihara and M. Ozawa, Accidental surfaces in knot complements, J. Knot Theory and its Ramifications, 9 (2000), 725-733. MR 2001f:57007
- W. Jaco, Lectures on Three Manifold Topology, AMS Conference board of Math. No. 43, 1980. MR 81k:57009
- M. Kouno, K. Motegi and T. Shibuya, Twisting and knot types, J. Math. Soc. Japan 44 (1992), 199-216. MR 93e:57011
- H. C. Lyon, Incompressible surfaces in knot spaces, Trans. Amer. math. soc, 157 (1971), 53-62. MR 43:1169
- W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1978.
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
Retrieve articles in all journals
with MSC (2000):
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
Department of Mathematics, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan
Received by editor(s):
July 25, 2000
Received by editor(s) in revised form:
July 25, 2002
April 30, 2003
The first author was supported in part by Fellowship of the Japan Society for the Promotion of Science for Japanese Junior Scientists
Ronald A. Fintushel
© Copyright 2003
American Mathematical Society