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Knot theory and the Casson invariant in Artin presentation theory

Abstract

In Artin presentation theory, a smooth, compact four-manifold is determined by a certain type of presentation of the fundamental group of its boundary. Topological invariants of both three-and four-manifolds can be calculated solely in terms of functions of the discrete Artin presentation. González-Acuña proposed such a formula for the Rokhlin invariant of an integral homology three-sphere. This paper provides a formula for the Casson invariant of rational homology spheres. Thus, all 3D Seiberg-Witten invariants can be calculated by using methods of the theory of groups in Artin presentation theory. The Casson invariant is closely related to canonical knots determined by an Artin presentation. It is also shown that any knot in any three-manifold appears as a canonical knot in Artin presentation theory. An open problem is to determine 4D Seiberg-Witten and Donaldson invariants in Artin presentation theory.

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References

  1. 1.

    S. Akbulut and J. McCarthy, Casson’s Invariant for Oriented Homology 3-Spheres, Math. Notes, Vol. 36, Princeton Univ. Press (1990).

  2. 2.

    L. Armas-Sanabria, “The µ invariant and the Casson invariant of 3-manifolds obtained by surgery on closed pure 3-braids,” J. Knot Theory Ramifications, 13, No. 3, 427–440 (2004).

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    J. S. Calcut, Torelli Actions and Smooth Structures on 4-Manifolds, Ph.D. Thesis, University of Maryland (2004).

  4. 4.

    J. S. Calcut and H. E. Winkelnkemper, “Artin presentations of complex surfaces,” accepted at Bol. Soc. Mat. Mexicana (volume dedicated to F. González-Acuña).

  5. 5.

    R. Fintushel and R. Stern, “Knots, links, and 4-manifolds,” Invent. Math., 134, No. 2, 363–400 (1998).

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    R. Gompf and A. Stipsicz, 4--Manifolds and Kirby Calculus, Grad. Stud. Math., Vol. 20, Amer. Math. Soc. (1999).

  7. 7.

    F. González-Acuña, Open Books, University of Iowa (1975).

  8. 8.

    W. Lickorish, “A representation of orientable combinatorial 3-manifolds,” Ann. Math., 76, No. 3, 531–540 (1962).

    Article  MathSciNet  Google Scholar 

  9. 9.

    W. Lickorish, “A finite set of generators for the homeotopy group of a 2-manifold,” Math. Proc. Cambridge Philos. Soc., 60, 769–778 (1964).

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    W. Lickorish, “A foliation for 3-manifolds,” Ann. Math., 82, 414–420 (1965).

    Article  MathSciNet  Google Scholar 

  11. 11.

    Y. Lim, “The equivalence of Seiberg-Witten and Casson invariants for homology 3-spheres,” Math. Res. Lett., 6, 631–643 (1999).

    MATH  MathSciNet  Google Scholar 

  12. 12.

    G. Meng and C. H. Taubes, “SW = Milnor torsion,” Math. Res. Lett., 3, No. 5, 661–674 (1996).

    MATH  MathSciNet  Google Scholar 

  13. 13.

    D. Rolfsen, Knots and Links, Publish or Perish, Berkeley (1976).

    MATH  Google Scholar 

  14. 14.

    K. Walker, An Extension of Casson’s Invariant, Ann. Math. Stud., Vol. 126, Princeton Univ. Press (1992).

  15. 15.

    H. E. Winkelnkemper, “Artin presentations. I: Gauge theory, 3 + 1 TQFTs and the braid groups,” J. Knot Theory Ramifications, 11, No. 2, 223–275 (2002).

    MATH  Article  MathSciNet  Google Scholar 

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 119–126, 2005.

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Calcut, J.S. Knot theory and the Casson invariant in Artin presentation theory. J Math Sci 144, 4446–4450 (2007). https://doi.org/10.1007/s10958-007-0283-2

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Keywords

  • Fundamental Group
  • Braid Group
  • Dehn Twist
  • Rational Homology
  • Integral Homology