Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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KNOTS IN S3 ADMITTING GRAPH MANIFOLD DEHN SURGERIES

  • Kang, Sungmo (Department of Mathematics Education Chonnam National University)
  • Received : 2014.03.10
  • Published : 2014.11.01
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Abstract

In this paper, we construct infinite families of knots in $S^3$ which admit Dehn surgery producing a graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. In particular, we show that for any natural numbers a, b, c, and d with $a{\geq}3$ and $b,c,d{\geq}2$, there are knots in $S^3$ admitting a graph manifold Dehn surgery consisting of two Seifert-fibered spaces over the disk with two exceptional fibers of indexes a, b, and c, d, respectively.

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References

  1. J. Berge, Some Knots with Surgeries Yielding Lens Spaces, Unpublished manuscript. Univ. of Texas at Austin, 1990.
  2. J. Berge, A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody, arXiv:0910.3038.
  3. J. Berge and S. Kang, The hyperbolic P=P, $P=SF_{d}$, and $P=SF_{m}$ knots in $S^{3}$, preprint.
  4. J. Berge and S. Kang, Primitive and Seifert curves in a genus two handlebody, preprint.
  5. J. Dean, Hyperbolic knots with small Seifert-fibered Dehn surgeries, Ph.D. thesis, University of Texas at Austin, 1996.
  6. J. Dean, Small Seifert- bered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003), 435-472. https://doi.org/10.2140/agt.2003.3.435
  7. M. Eudave-Munoz, Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc. 330 (1992), no. 2, 463-501. https://doi.org/10.1090/S0002-9947-1992-1112545-X
  8. M. Eudave-Munoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Proceedings of the 1993 International Georgia Topology Conference, W. Kazea (Ed.), 35-61, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997.
  9. S. Kang, Knots admitting Seifert-fibered surgeries over $S^{2}$ with four exceptional fibers, to appear in Bull. Korean Math. Soc.
  10. R. P. Osborne and R. S. Stevens, Group presentations corresponding to spines of 3-manifolds II, Trans. Amer. Math. Soc. 234 (1977), no. 1, 213-243.
  11. D. Rolfsen, Knots and Links, Math. Lect. Ser. 7, Publish or Perish, Berkeley, California, 1976.
  12. M. Teragaito, Hyperbolic knots with three toroidal Dehn surgeries, J. Knot Theory Rami fications 17 (2008), no. 9, 1051-1061. https://doi.org/10.1142/S0218216508006543

Cited by

  1. TWISTED TORUS KNOTS WITH GRAPH MANIFOLD DEHN SURGERIES vol.53, pp.1, 2016, https://doi.org/10.4134/BKMS.2016.53.1.273