Closed incompressible surfaces

in knot complements

Authors:
Elizabeth Finkelstein and Yoav Moriah

Journal:
Trans. Amer. Math. Soc. **352** (2000), 655-677

MSC (1991):
Primary 57M25, 57M99, 57N10

DOI:
https://doi.org/10.1090/S0002-9947-99-02233-3

Published electronically:
September 9, 1999

MathSciNet review:
1487613

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show that given a knot or link in a -plat projection with and , where is the length of the plat, if the twist coefficients all satisfy then has at least nonisotopic essential meridional planar surfaces. In particular if is a knot then contains closed incompressible surfaces. In this case the closed surfaces remain incompressible after all surgeries except perhaps along a ray of surgery coefficients in .

**[BZ]**G. Burde and H. Zieschang, Knots, De Gruyter Studies in Mathematics, 5, New York, 1985. MR**87b:57004****[CGLS]**M. Culler, C. Gordon, J. Luecke, and P. Shalen,*Dehn surgery on knots*, Ann. of Math.**125**(1987), 237-300. MR**88a:57026****[CL]**D. Cooper and D. Long,*Derivative varieties and the pure braid group*, Amer. J. Math.**115**(1993), 137-160. MR**94b:57003****[Fi]**E. Finkelstein,*Closed incompressible surfaces in closed braid complements*, J. Knot Theory Ramifications**7**(1998), 335-379. CMP**98:13****[GL]**C. McA. Gordon and J. Luecke,*Reducible manifolds and Dehn surgery*, Topology**35**(1996), 385-409. MR**97b:57013****[GR]**C. McA. Gordon and A. Reid,*Tangle decompositions of tunnel number one knots and links*, J. Knots Theory Ramifications**4**(1995), 389-409. MR**96m:57016****[He]**J. Hempel,*-manifolds*, Ann. of Math. Studies 86, Princeton University Press, Princeton, N.J., 1976. MR**54:3702****[JS]**W. Jaco and P. Shalen,*Seifert fibered spaces in -manifolds*, Mem. Amer. Math. Soc.**21**(1979). MR**81c:57010****[HK]**D. Heath and T. Kobayashi,*A search method for a thin position of a link*, preprint.**[LM]**M. Lustig and Y. Moriah,*Generalized Montesinos knots, tunnels and -torsion*, Math. Ann.**295**(1993), 167-189. MR**94b:57011****[LP]**M. T. Lozano and J. H. Przytycki,*Incompressible surfaces in the exterior of a closed -braid*, Math. Proc. Cambridge Philos. Soc.**98**(1985), 275-299. MR**87a:57013****[Ly]**H. Lyon,*Incompressible surfaces in knot spaces*, Trans. Amer. Math. Soc.**157**(1971), 53-62. MR**43:1169****[Me]**W. Mensco,*Closed incompressible surfaces in alternating knot and link complements*, Topology**23**(1984), 37-44. MR**86b:57004****[Oe]**U. Oertel,*Closed incompressible surfaces in complements of star links*, Pacific J. Math.**111**(1984), 209-230. MR**85j:57008****[Sh]**H. Short,*Some closed incompressible surfaces in knot complements which survive surgery*, London Math. Soc. Lecture Notes Ser.,**95**(1985), 179-194. MR**88d:57006****[Sw]**G. A. Swarup,*On incompressible surfaces in the complements of knots*, J. Indian Math. Soc.**37**(1973), 9-24. MR**50:14757****[Th]**A. Thompson,*Thin position and bridge number for knots in the -sphere*, Topology**36**(1997), 505-507. MR**97m:57013****[Wu 1]**Y. Q. Wu,*Incompressibility of surfaces in surgered -manifolds*, Topology**31**(1992), 271-279. MR**94e:57027****[Wu 2]**Y. Q. Wu,*The classification of nonsimple algebraic tangles*, Math. Ann.**304**(1996), 457-480. MR**97b:57010**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
57M25,
57M99,
57N10

Retrieve articles in all journals with MSC (1991): 57M25, 57M99, 57N10

Additional Information

**Elizabeth Finkelstein**

Affiliation:
Department of Mathematics, (CUNY) Hunter College, New York, New York 10021

Email:
efinkels@shiva.hunter.cuny.edu

**Yoav Moriah**

Affiliation:
Department of Mathematics, Technion, Haifa 32000, Israel

Email:
ymoriah@techunix.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-99-02233-3

Received by editor(s):
May 23, 1996

Received by editor(s) in revised form:
October 10, 1997

Published electronically:
September 9, 1999

Article copyright:
© Copyright 1999
American Mathematical Society