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Characteristic Genera of Closed Orientable 3-Manifolds

  • KAWAUCHI, AKIO
  • Received : 2015.08.13
  • Accepted : 2015.11.13
  • Published : 2015.12.23

Abstract

A complete invariant defined for (closed connected orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself can be reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In a previous work, a characteristic lattice point invariant defined for the 3-manifolds was constructed by using an embedding of the prime links into the set of lattice points. In this paper, a characteristic rational invariant defined for the 3-manifolds called the characteristic genus defined for the 3-manifolds is constructed by using an embedding of a set of lattice points called the PDelta set into the set of rational numbers. The characteristic genus defined for the 3-manifolds is also compared with the Heegaard genus, the bridge genus and the braid genus defined for the 3-manifolds. By using this characteristic rational invariant defined for the 3-manifolds, a smooth real function with the definition interval (-1, 1) called the characteristic genus function is constructed as a characteristic invariant defined for the 3-manifolds.

Keywords

Braid;3-manifold;Prime link;Characteristic genus;Characteristic function

References

  1. J. S. Birman, Braids, links, and mapping class groups, Ann. Math. Studies, 82(1974), Princeton Univ. Press.
  2. W. Blaschke, Eine Erweiterung des Satzes von Vitali uber Folgen analytischer Funktionen, Berichte Math.-Phys. Kl., Sachs. Gesell. der Wiss. Leipzig, 67(1915), 194-200.
  3. A. Kawauchi, A survey of knot theory, (1996), Birkhauser.
  4. A. Kawauchi, Topological imitation of a colored link with the same Dehn surgery manifold, in: Proceedings of Topology in Matsue 2002, Topology Appl., 146-147(2005), 67-82.
  5. A. Kawauchi, A tabulation of 3-manifolds via Dehn surgery, Boletin de la Sociedad Matematica Mexicana (3), 10(2004), 279-304.
  6. A. Kawauchi and I. Tayama, Enumerating the prime knots and links by a canonical order, in: Proc. 1st East Asian School of Knots, Links, and Related Topics (Seoul, Jan. 2004), (2004), 307-316.
  7. A. Kawauchi and I. Tayama, Enumerating the exteriors of prime links by a canonical order, in: Proc. Second East Asian School of Knots, Links, and Related Topics in Geometric Topology (Darlian, Aug. 2005), (2005), 269-277.
  8. A. Kawauchi and I. Tayama, Enumerating prime links by a canonical order, Journal of Knot Theory and Its Ramifications, 15(2006), 217-237. https://doi.org/10.1142/S0218216506004439
  9. A. Kawauchi and I. Tayama, Enumerating 3-manifolds by a canonical order, Intelligence of low dimensional topology 2006, Series on Knots and Everything, 40(2007), 165-172.
  10. A. Kawauchi and I. Tayama, Enumerating prime link exteriors with lengths up to 10 by a canonical order, Proceedings of the joint conference of Intelligence of Low Dimensional Topology 2008 and the Extended KOOK Seminar (Osaka, Oct. 2008), (2008), 135-143.
  11. A. Kawauchi and I. Tayama, Enumerating homology spheres with lengths up to 10 by a canonical order, Proceedings of Intelligence of Low-Dimensional Topology 2009 in honor of Professor Kunio Murasugi's 80th birthday (Osaka, Nov. 2009), (2009), 83-92.
  12. A. Kawauchi and I. Tayama, Enumerating 3-manifolds with lengths up to 9 by a canonical order, Topology Appl., 157(2010), 261-268. https://doi.org/10.1016/j.topol.2009.04.028
  13. A. Kawauchi and I. Tayama, Representing 3-manifolds in the complex number plane, preprint. (http://www.sci.osaka-cu.ac.jp/-kawauchi/index.htm)
  14. A. Kawauchi, I. Tayama and B. Burton, Tabulation of 3-manifolds of lengths up to 10, Proceedings of International Conference on Topology and Geometry 2013, joint with the 6th Japan-Mexico Topology Symposium, Topology and its Applications (to appear). http://dx.doi.org/10.1016/j.topol.2015.05.036 https://doi.org/10.1016/j.topol.2015.05.036
  15. B. von Kerekjarto, Vorlesungen uber Topologie, Spinger, Berlin, 1923.
  16. R. Kirby, A calculus for framed links in $S^3$, Invent. Math., 45(1978), 35-56. https://doi.org/10.1007/BF01406222
  17. J. Milnor and W. Thurston, Characteristic numbers of 3-manifolds, Enseignment Math., 23(1977), 249-254.
  18. Y. Nakagawa, A family of integer-valued complete invariants of oriented knot types, J. Knot Theory Ramifications, 10(2001), 1160-1199.
  19. S. Okazaki, On Heegaard genus, bridge genus and braid genus for a 3-manifold, J. Knot Theory Ramifications, 20(2011), 1217-1227. https://doi.org/10.1142/S0218216511009145
  20. D. Rolfsen, Knots and links, (1976), Publish or Perish.

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