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THE DISJOINT CURVE PROPERTY AND BRIDGE SURFACES

  • Published : 2009.09.30

Abstract

We show that every bridge surface of certain types of (1, 1) prime knot has the disjoint curve property. Also we determine when a bridge surface of a pretzel knot of type (.2, 3, n) has the disjoint curve property.

Keywords

References

  1. C. Adams, The Knot Book, W. H. Freeman and Company, New York, 1994
  2. C. Adams, J. Brock, J. Bugbee, and et al, Almost alternating links, Topology Appl. 46 (1992), no. 2, 151–165 https://doi.org/10.1016/0166-8641(92)90130-R
  3. A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275–283 https://doi.org/10.1016/0166-8641(87)90092-7
  4. W. Haken, Studies in Modern Topology; Some results on surfaces in 3-manifolds, Prentice Hall, 1968
  5. C. Hayashi, 1-genus 1-bridge splittings for knots, Osaka J. Math. 41 (2004), no. 2, 371–426
  6. C. Hayashi, Genus one 1-bridge positions for the trivial knot and cabled knots, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 1, 53–65
  7. C. Hayashi, Satellite knots in 1-genus 1-bridge positions, Osaka J. Math. 36 (1999), no. 3, 711–729
  8. J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631–657
  9. J. Hoste, M. Thistlethwaite, and J.Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998), no. 4, 33–48
  10. D. Kim and E. Lee, Some invariants of pretzel links, arXiv:math.GT0704.1432v1, preprint
  11. T. Kobayashi, Heights of simple loops and pseudo-Anosov homeomorphisms, Braids (Santa Cruz, CA, 1986), 327–338, Contemp. Math., 78, Amer. Math. Soc., Providence, RI, 1988
  12. W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44
  13. K. Morimoto, On minimum genus Heegaard splittings of some orientable closed 3- manifolds, Tokyo J. Math. 12 (1989), no. 2, 321–355
  14. D. Rolfsen, Knots and Links, Publish or Perish, Inc., Houston, TX, 1990
  15. T. Saito, Genus one 1-bridge knots as viewed from the curve complex, Osaka J. Math. 41 (2004), no. 2, 427–454
  16. T. Saito, Disjoint pairs of annuli and disks for Heegaard splittings, J. Korean Math. Soc. 42 (2005), no. 4, 773–793 https://doi.org/10.4134/JKMS.2005.42.4.773
  17. A. Thompson, The disjoint curve property and genus 2 manifolds, Topology Appl. 97 (1999), no. 3, 273–279 https://doi.org/10.1016/S0166-8641(98)00063-7