Dehn surgery on the figure 8 knot: Immersed surfaces
Authors:
I. R. Aitchison, S. Matsumoto and J. H. Rubinstein
Journal:
Proc. Amer. Math. Soc. 127 (1999), 24372442
MSC (1991):
Primary 57Q35; Secondary 57M50
Published electronically:
March 24, 1999
MathSciNet review:
1485454
Fulltext PDF Free Access
Abstract 
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Abstract: It is known that about 70% of surgeries on the figure 8 knot give manifolds which contain immersed incompressible surfaces. We improve this to about 80% by giving a very simple proof that all even surgeries give manifolds containing such a surface. Moreover, we give a quick proof that every surgery is virtually Haken, thereby partially dealing with some exceptional cases in Baker's results.
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 [ALR]
 I.R. Aitchison, E. Lumsden and J.H. Rubinstein, Cusp structure of alternating links, Invent. Math. 109 (1992), 473494. MR 93h:57007
 [AMR1]
 I. R. Aitchison, S. Matsumoto, and J. H. Rubinstein, Immersed surfaces in cubed manifolds, Asian J. Math 1 (1997), 8595. CMP 98:04
 [AMR2]
 , Immersed surfaces in the figure8 knot complement, preprint, 1996.
 [AR]
 I. R. Aitchison and J. H. Rubinstein, Combinatorial cubings, cusps and the dodecahedral knots, in: Proc. of the Special Semester on Topology at Ohio State University, 1990 (de Gruyter, Berlin  New York, 1992) pp. 1726. MR 93i:57016
 [Ba]
 M. D. Baker, On coverings of figure eight knot surgeries, Pacific J. Math. 150 (1991), 215228. MR 92g:57004
 [Bu]
 G. Burde, On branched coverings of , Canad. J. Math. 23 (1971), 8489. MR 43:6908
 [Fo1]
 R. H. Fox, Construction of simply connected 3manifolds, in: Topology of 3manifolds and related topics, Proc. The Univ. of Georgia Institute, 1961 (PrenticeHall, Englewood Cliffs, N.J., 1962) pp. 213216. MR 25:3539
 [Fo2]
 R. H. Fox, Metacyclic invariants of knots and links, Canad. J. Math. 22 (1970), 193201. MR 41:6197
 [HM]
 J. Hass and W. Menasco, Topologically rigid nonHaken 3manifolds, J. Austral. Math. Soc. (Series A) 55 (1993), 6071. MR 94g:57017
 [Ha]
 A. Hatcher, Hyperbolic structures of arithmetic type on some link complements, J. London Math. Soc. (2) 27 (1983), 345355. MR 84m:57005
 [He1]
 J. Hempel, Coverings of Dehn fillings of surface bundles, Topology Appl. 24 (1986), 157170. MR 88h:57011
 [Ki]
 R.C. Kirby (ed.), Problems in lowdimensional topology, in: Geometric Topology (2 vols), 1993 Georgia International Topology Conference Proceedings (ed. William H. Kazez) (American Mathematical Society, Providence, 1997). CMP 98:01
 [KL]
 S. Kojima and D. Long, Virtual Betti numbers of some hyperbolic 3manifolds, in: A fete of topology: Papers dedicated to Itiro Tamura (Academic Press, New York, 1988), pp.417437. MR 89a:57060
 [Mo]
 S. Morita, Finite coverings of punctured torus bundles and the first Betti number, Sci. Papers College Arts Sci. Univ. Tokyo 35 (1986), no. 2, 109121. MR 88b:57021
 [NR]
 W. D. Neumann and A. W. Reid, Arithmetic of hyperbolic manifolds, in: Proc. of the Special Semester on Topology at Ohio State University, 1990 (de Gruyter, Berlin  New York, 1992) pp. 273  310. MR 94c:57024
 [N]
 A. Nicas, An infinite family of nonHaken hyperbolic 3manifolds with vanishing Whitehead groups, Math. Proc. Camb. Phil. Soc. 99 (1986), 239246. MR 87h:57018
 [Pr]
 J. H. Przytycki, Incompressibility of surfaces after Dehn surgery, Michigan Math. J. 30 (1983), 289308. MR 86g:57012
 [Re]
 A. W. Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991), 171184. MR 92a:57011
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 D. Rolfsen, Knots and Links (Publish or Perish, Inc., Houston, 1990). MR 95c:57018
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Additional Information
I. R. Aitchison
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
Email:
iain@maths.mu.oz.au
S. Matsumoto
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
Email:
saburo@is.titech.ac.jp
J. H. Rubinstein
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
Email:
rubin@maths.mu.oz.au
DOI:
http://dx.doi.org/10.1090/S0002993999047164
PII:
S 00029939(99)047164
Keywords:
Cubed manifolds,
immersed incompressible surfaces,
dihedral cover,
Dehn surgery
Received by editor(s):
November 12, 1996
Received by editor(s) in revised form:
October 21, 1997
Published electronically:
March 24, 1999
Additional Notes:
This research was partially supported by the Australian Research Council.
Communicated by:
Ronald A. Fintushel
Article copyright:
© Copyright 1999
American Mathematical Society
