HOMOLOGY 3-SPHERES OBTAINED BY SURGERY ON EVEN NET DIAGRAMS

Title & Authors
HOMOLOGY 3-SPHERES OBTAINED BY SURGERY ON EVEN NET DIAGRAMS
Lee, Sang-Youl;

Abstract
In this paper, we characterize surgery presentations for $\small{\mathbb{Z}}$-homology 3-spheres and $\small{\mathbb{Z}/2\mathbb{Z}}$-homology 3-spheres obtained from $\small{S^3}$ by Dehn surgery along a knot or link which admits an even net diagram and show that the Casson invariant for $\small{\mathbb{Z}}$-homology spheres and the $\small{{\mu}}$-invariant for $\small{\mathbb{Z}/2\mathbb{Z}}$-homology spheres can be directly read from the net diagram. We also construct oriented 4-manifolds bounding such homology spheres and find their some properties.
Keywords
Casson invariant;Dehn surgery;homology 3-sphere;$\small{{\mu}}$-invariants;net diagram;4-manifold;
Language
English
Cited by
References
1.
S. Akbulut and J. McCathy, Casson's Invariant for Oriented Homology 3-Spheres, Princeton Math. Notes 36, Princeton University Press, 1990.

2.
J. Eells and N. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93-110.

3.
R. A. Fenn and C. Rourke, On Kirby's calculus of links, Topology 18 (1979), no. 1, 1-15.

4.
H. M. Hilden, Representations of homology 3-spheres, Pacific J. Math. 94 (1981), no. 1, 125-129.

5.
F. Hirzebruch, W. Neumann, and S. Koh, Differentiable Manifolds and Quadratic Forms, Marcel Dekker, 1971.

6.
R. Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978), no. 1, 35-56.

7.
S. Y. Lee and M. Seo, Formulas for the Casson invariant of certain integral homology 3-spheres, J. Knot Theory Ramifications 18 (2009), no. 11, 1551-1576.

8.
W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. Math. 76 (1962), 531-540.

9.
S. Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987), no. 2, 268-278, 345.

10.
V. Rokhlin, New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk SSSR 84 (1952), 221-224.

11.
D. Rolfsen, Rational surgery calculus: extension of Kirby's theorem, Pacific J. Math. 110 (1984), no. 2, 377-386.

12.
D. Rolfsen, Knots and Links, Publish or Perish, 1990.

13.
N. Saveliev, Lectures on the Topology of 3-Manifolds, de Gruyter, 1999.

14.
N. Saveliev, Invariants for Homology 3-Spheres, Springer, 2002.

15.
A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503-528.