Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T14:55:52.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Discriminants of Casson–Gordon invariants

Published online by Cambridge University Press:  24 October 2008

Patrick Gilmer  and
Charles Livingston
Patrick Gilmer
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
Charles Livingston
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Casson–Gordon invariants were first used to prove that certain algebraically slice knots in S3 are not slice knots [2, 3]. Since then they have been applied to a wide range of problems, including embedding problems and questions relating to boundary links [2, 10, 21, 25]. The most general Casson–Gordon invariant takes its value in L0(ℚ(ζd)(t)) ⊗ ℚ; here ζd denotes a primitive dth root of unity. Litherland [20] observed that one could usually tensor with ℤ(2) instead of ℚ, and in this way preserve the 2-torsion in the Witt group. He then constructed new examples of non-slice genus two knots which were detected with torsion classes in L0(ℚ(ζd)) ⊗ ℤ(2) modulo the image of L0(ℚ(ζd)) ⊗ ℤ(2).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 1 , July 1992 , pp. 127 - 139
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alexander, J. P., Conner, P. E. and Hamrick, G. C.. Odd Order Group Actions and Witt Classification of Innerproducts. Lecture Notes in Math. Vol. 625 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[2]Casson, A. and Gordon, C. McA.. Cobordism of classical knots. In A la Recherche de la Topologie Perdue (editors Mann, A. and Guillon, L.). Progress in Mathematics no. 62 (Birkhauser, 1986), pp. 181–197.Google Scholar
[3]Casson, A. and Gordon, C. McA.. On slice knots in dimension three. Proc. Symp. Pure Math. 30 (1978), 39–53.CrossRefGoogle Scholar
[4]Cochran, T. and Orr, K.. Not all links are concordant to boundary links. Bull. Amer. Math. Soc. 23 (1990), 99–106.CrossRefGoogle Scholar
[5]Conner, P.. Notes on the Witt Classification of Iziermitian Innerproduct Spaces over a Ring of Algebraic Integers (University of Texas Press, 1979).Google Scholar
[6]Gilmer, P.. Configurations of surfaces in 4-manifolds. Trans. Amer. Math. Soc. 264 (1981), 353–380.Google Scholar
[7]Gilmer, P.. Slice knots in S 3. Quart. J. Math. Oxford Ser. (2) 34 (1983), 305–322.CrossRefGoogle Scholar
[8]Gilmer, P.. Classical knot and link concordance. Preprint.Google Scholar
[9]Gilmer, P. and Livingston, C.. On embedding 3-manifolds in 4-space. Topology 22 (1983), 241–252.CrossRefGoogle Scholar
[10]Gilmer, P. and Livingston, C.. The Casson–Gordon invariant and link concordance. Topology, to appear.Google Scholar
[11]Gordon, C. McA.. A short proof of a theorem of Plans on the homology of the branched cyclic coverings of a knot. Bull. Amer. Math. Soc. 77 (1971), 85–87.CrossRefGoogle Scholar
[12]Gordon, C. McA.. Aspects of classical knot theory. In Knot Theory, Lecture Notes in Math. Vol. 685 (Springer-Verlag, 1978) pp. 1–60.CrossRefGoogle Scholar
[13]Gordon, C. McA.. Knots, homology spheres, and contractible 4-manifolds. Topology 14 (1975), 151–172.CrossRefGoogle Scholar
[14]Hirzebruch, F., Neumann, W. and Koh, S.. Differentiable Manifolds and Quadratic Forms (Marcel Dekker, 1971).Google Scholar
[15]Hirzebruch, F. and Zagier, B.. The Atiyah–Singer Theorem and Elementary Number Theory (Publish or Perish, 1974).Google Scholar
[16]Kauffman, L. H.. The Conway polynomial. Topology 20 (1981), 101–108.CrossRefGoogle Scholar
[17]Kirby, R.. A calculus for framed links in S 3. Invent. Math. 45 (1978), 35–56.CrossRefGoogle Scholar
[18]Levine, J.. Knot cobordism groups in codimension two. Comment. Math. Helv. 44 (1969), 229–244.CrossRefGoogle Scholar
[19]Levine, J.. Invariants of knot cobordism. Invent. Math. 8 (1969), 98–110.CrossRefGoogle Scholar
[20]Litherland, R.. A formula for the Casson–Gordon invariants of a knot. Preprint.Google Scholar
[21]Livingston, C.. Links not concordant to boundary links. Proc. Amer. Math. Soc. 110 (1990), 1129–1131.CrossRefGoogle Scholar
[22]Marcus, D. A.. Number Fields (Springer-Verlag, 1977).CrossRefGoogle Scholar
[23]Matic, G.. SO(3)-connections and rational homology cobordisms. J. Differential Geometry 28 (1988), 277–307.CrossRefGoogle Scholar
[24]Plans, A.. Contribution to the study of the homology groups of the cyclic ramified coverings corresponding to a knot. Rev. Acad. Cienc. Madrid 47 (1953), 161–193.Google Scholar
[25]Ruberman, B.. Doubly slice knots and the Casson–Gordon invariants. Trans. Amer. Math. Soc. 279 (1983), 569–588.CrossRefGoogle Scholar
[26]Ruberman, D.. Rational homology cobodisms of rational space forms. Topology 27 (1988), 401–414.CrossRefGoogle Scholar
[27]Seifert, H.. Über das Geschlecht von Knoten. Math. Ann. 110 (1934), 571–592.CrossRefGoogle Scholar