Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-12T20:42:29.801Z Has data issue: false hasContentIssue false
  • English
  • Français

Signatures of Covering Links

Published online by Cambridge University Press:  20 November 2018

C. Mca. Gordon ,
R. A. Litherland  and
K. Murasugi
C. Mca. Gordon
Affiliation:
University of Texas, at Austin, Austin, Texas
R. A. Litherland
Affiliation:
University of Cambridge, Cambridge, England
K. Murasugi
Affiliation:
University of Toronto, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A (tame) knot kn in S3 is said to have period n if there exists a homeomorphism ϕ: S3 → S3, necessarily orientationpreserving, such that

  • (i) the fixed point set of ϕ is a circle disjoint from kn

  • (ii) ϕ(kn) = kn;

  • (iii) ϕ has order n.

Several necessary conditions for a knot to have period n have already been established in the literature; see [3] [11] [14] [18]. Here we establish the following further condition, involving the signature σ(kn) of (kn).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 381 - 394
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Atiyah, M. F. and Singer, I. M., The index of elliptic operators, III, Ann. of Math. 87 (1968), 546–604.Google Scholar
2. Brieskorn, E., Beispiele zur Differentialtopologie von Singularitaten, Invent. Math. 2 (1966), 1–14.Google Scholar
3. Burde, G., Ûber periodische Knoten, Archiv der Math. 30 (1978), 487–492.Google Scholar
4. Conner, P. E., Transformation groups on a K(ir, 1), / / , Michigan Math. J. 6 (1959), 413–417.Google Scholar
5. Goldsmith, D. L., Symmetric fibered links in Knots, groups and 3-manifolds: papers dedicated to the memory of R. H. Fox, Ann. of Math. Studies 84 (Princeton Univ. Press, Princeton N.J., 1975), 3–25.Google Scholar
6. Gordon, C. McA. and Litherland, R. A., On a theorem of Murasugi, Pacific J. Math. 82 (1979), 69–74.Google Scholar
7. Hirzebruch, F., Singularities and exotic spheres, Sém. Bourbaki 314 (1966/67).Google Scholar
8. Hosokawa, F. and Kinoshita, S., On the homology group of branched cyclic covering spaces of links, Osaka Math. J. 12 (1960), 331–355.Google Scholar
9. Levine, J., Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244.Google Scholar
10. Litherland, R. A., Topics in knot theory, thesis, Cambridge University (1978).Google Scholar
11. Liidicke, U., Darstellungen der Verkettungsgruppe und zyklische Knoten, Dissertation, Frankfurt am Main (1978).Google Scholar
12. Milnor, J., On the 3-dimensional Brieskorn manifolds M(p, a, r) in Knots, groups and 3-manifolds: papers dedicated to the memory of R. H. Fox, Ann. of Math. Studies 84 (Princeton Univ. Press, Princeton N.J., 1975), 175–225.Google Scholar
13. Murasugi, K., On the signature of links, Topology 9 (1970), 283–298.Google Scholar
14. Murasugi, K., On periodic knots, Comment. Math. Helv. 46 (1971), 162–174.Google Scholar
15. Murasugi, K., On closed 3-braids, Memoirs of the Amer. Math. Soc. 151 (Amer. Math. Soc, Providence R.I., 1974).Google Scholar
16. Riley, R., Automorphisms of excellent link groups, to appear.Google Scholar
17. Tristram, A. G., Some cobordism invariants for links, Proc. Camb. Phil. Soc. 66 (1969), 251–264.Google Scholar
18. Trotter, H. F., Periodic automorphisms of groups and knots, Duke Math. J. 28 (1961), 553–558.Google Scholar
19. Viro, O. Ja., Branched coverings of manifolds with boundary and link invariants, I, Math. USSR Izvestija 7 (1973), 1239–1256.Google Scholar
20. Waldhausen, F., Irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 55–88.Google Scholar