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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: [email protected]
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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

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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