Abstract:We generalize a theorem of Finkelstein and Moriah and show that if a link $L$ has a $2n$-plat projection satisfying certain conditions, then its complement contains some closed essential surfaces. In most cases these surfaces remain essential after any totally nontrivial surgery on $L$.
- Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088
- Elizabeth Finkelstein and Yoav Moriah, Closed incompressible surfaces in knot complements, Trans. Amer. Math. Soc. 352 (2000), no. 2, 655–677. MR 1487613, DOI 10.1090/S0002-9947-99-02233-3
- —, Tubed incompressible surfaces in knot and link complements, Topology Appl. 96 (1999), 153–170.
- A. Hatcher and W. Thurston, Incompressible surfaces in $2$-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246. MR 778125, DOI 10.1007/BF01388971
- W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44. MR 721450, DOI 10.1016/0040-9383(84)90023-5
- Martin Scharlemann, Producing reducible $3$-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481–500. MR 1071370, DOI 10.1016/0040-9383(90)90017-E
- Ying-Qing Wu, The classification of nonsimple algebraic tangles, Math. Ann. 304 (1996), no. 3, 457–480. MR 1375620, DOI 10.1007/BF01446301
- Ying-Qing Wu
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
- Email: firstname.lastname@example.org
- Received by editor(s): February 22, 2000
- Received by editor(s) in revised form: March 27, 2000
- Published electronically: April 2, 2001
- Additional Notes: The author was supported in part by NSF grant #DMS 9802558.
- Communicated by: Ronald A. Fintushel
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 3417-3423
- MSC (1991): Primary 57N10, 57M25
- DOI: https://doi.org/10.1090/S0002-9939-01-05938-X
- MathSciNet review: 1845021