# Characteristic Genera of Closed Orientable 3-Manifolds

• KAWAUCHI, AKIO
• Accepted : 2015.11.13
• Published : 2015.12.23
• 19 4

#### 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.

#### 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., Sachs. Gesell. der Wiss. Leipzig, 67(1915), 194-200.
3. A. Kawauchi, A survey of knot theory, (1996), Birkhauser.
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 Kerekjarto, 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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