Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-29T20:44:58.924Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

Published online by Cambridge University Press:  10 April 2007

YUYA KODA
YUYA KODA*
Affiliation:
Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan. e-mail: koda@math.keio.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Let K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 2 , March 2007 , pp. 259 - 268
Copyright
Copyright © Cambridge Philosophical Society 2007

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] Burde, G. and Zieschang, H.. Knots. De Gruyter Studies in Math. vol. 5 (Walter de Gruyter & Co., 1985).Google Scholar
[2] Cattabriga, A.. The Alexander polynomial of (1, 1)-knots. J. Knot Theory Ramifications 15 (2006), 1119–1129.CrossRefGoogle Scholar
[3] Cristofori, P., Mulazzani, M. and Vesnin, A.. Strongly-cyclic branched coverings of knots via (g, 1)-decompositions. To appear in Acta Math. Hungar.Google Scholar
[4] Cattabriga, A. and Mulazzani, M.. Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups. Math. Proc. Camb. Phil. Soc. 135 (2003), 137–146.CrossRefGoogle Scholar
[5] Crowell, R. H. and Fox, R. H.. Introduction to Knot Theory. Graduate Texts in Math. vol. 57 (Springer-Verlag, 1963).Google Scholar
[6] Davis, P. J.. Circulant Matrices. Pure and Applied Mathematics; a Wiley-Interscience Series of Texts, Monographs and Tracts (John Wiley and Sons, 1979).Google Scholar
[7] Fox, R. H. and Torres, G.. Dual presentations of the groups of a knot. Ann. of Math. (2) 59 (1954), 211–218.Google Scholar
[8] Lickorish, W. B. R.. An Introduction to Knot Theory. Graduate Texts in Math. vol. 175 (Springer-Verlag, 1997).Google Scholar
[9] Milnor, J.. A duality theorem for Reidemeister torsion. Ann. of Math. 76 (1962), 137–147.CrossRefGoogle Scholar
[10] Minkus, J.. The branched cyclic coverings of 2 bridge knots and links. Mem. Amer. Math. Soc. 255 (1982), 1–68.Google Scholar
[11] Miyazaki, K.. Band-sums are ribbon concordant to the connected sum. Proc. Amer. Math. Soc. 126 (1998), 3401–3406.CrossRefGoogle Scholar
[12] Seifert, H.. Über das Geschlecht von Knoten. Math. Ann. 110 (1934), 571–592.CrossRefGoogle Scholar
[13] Turaev, V. G.. The Alexander polynomial of a three-dimensional manifold. Math. USSR Sb. 26 (1975), 313–329.CrossRefGoogle Scholar
[14] Turaev, V.. Introduction to Combinatorial Torsion. Lectures Math. ETH Zürich (Birkhäuserl, 2001).CrossRefGoogle Scholar