Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:44:09.475Z Has data issue: false hasContentIssue false
  • English
  • Français

THE GENUS OF PERIODIC LINKS WITH RATIONAL QUOTIENTS

Part of: Low-dimensional topology

Published online by Cambridge University Press:  13 March 2009

SANG YOUL LEE  and
MYOUNGSOO SEO
SANG YOUL LEE
Affiliation:
Department of Mathematics, Pusan National University, Pusan 609-735, Korea (email: sangyoul@pusan.ac.kr)
MYOUNGSOO SEO*
Affiliation:
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea (email: myseo@knu.ac.kr)
*
✉For correspondence; e-mail: myseo@knu.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper, we prove that if K is any periodic link in S3 whose quotient link is a 2-bridge link, then one half of the degree of the reduced Alexander polynomial, the minimal genus, the free genus and the canonical genus of K are all the same. We also give criteria to determine whether a given periodic link has a 2-bridge link quotient and some properties of this kind of periodic link.

Keywords

periodic linkgenus2-bridge linkhomogeneous link

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 273 - 284
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Cromwell, P. R., ‘Homogeneous links’, J. London Math. Soc. 39 (1989), 535–552.CrossRefGoogle Scholar
[2] Cromwell, P. R., Knots and Links (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
[3] Edmonds, A., ‘Least area Seifert surfaces and periodic knots’, Topol. Appl. 18 (1984), 109–113.CrossRefGoogle Scholar
[4] Frankel, F. and Pontrjagin, L., ‘Ein Knotensatz mit Anwendung auf die Dimensionstheorie’, Math. Ann. 102 (1930), 785–789.CrossRefGoogle Scholar
[5] Hilden, H. M., Lozano, M. T. and Montesinos-Amilibia, J. M., ‘On the character variety of periodic knots and links’, Math. Proc. Camb. Phil. Soc. 129 (2000), 477–490.CrossRefGoogle Scholar
[6] Jang, H. J., Lee, S. Y. and Seo, M., ‘Casson knot invariants of periodic knots with rational quotients’, J. Knot Theory Ramifications 16 (2007), 439–460.CrossRefGoogle Scholar
[7] Kawauchi, A., A Survey of Knot Theory (Birkhäuser, Basel, 1996).Google Scholar
[8] Lee, S. Y., Park, M.-S. and Seo, M., ‘The Seifert matrices of periodic links with rational quotients’, Kyungpook Math. J. 47 (2007), 295–309.Google Scholar
[9] Lee, S. Y. and Seo, M., ‘Recurrence formulas for the Alexander polynomials of 2-bridge links and their covering links’, J. Knot Theory Ramifications 15 (2006), 179–203.CrossRefGoogle Scholar
[10] Murasugi, K., ‘On the genus of the alternating knot I’, J. Math. Soc. Japan 10 (1958), 94–105.Google Scholar
[11] Murasugi, K., ‘On the genus of the alternating knot II’, J. Math. Soc. Japan 10 (1958), 235–248.Google Scholar
[12] Murasugi, K., ‘On a certain numerical invariant of link types’, Trans. Amer. Math. Soc. 117 (1965), 387–422.CrossRefGoogle Scholar
[13] Murasugi, K., ‘On periodic knots’, Comm. Math. Helv. 35 (1971), 529–537.Google Scholar
[14] Naik, S., ‘Periodicity, genera and Alexander polynomials of knots’, Pacific J. Math. 166 (1994), 357–371.CrossRefGoogle Scholar
[15] Nakamura, T., ‘On canonical genus of fibered knot’, J. Knot Theory Ramifications 11 (2002), 341–352.CrossRefGoogle Scholar
[16] Rolfsen, D., Knots and Links (AMS/Chelsea, New York, 2003).Google Scholar
[17] Seifert, H., ‘Über das Geschlecht von Knoten’, Math. Ann. 110 (1936), 571–592.CrossRefGoogle Scholar
[18] Shinohara, Y., ‘On the signature of knots and links’, Trans. Amer. Math. Soc. 156 (1971), 273–285.CrossRefGoogle Scholar
[19] Tripp, J. J., ‘The canonical genus of whitehead doubles of a family torus knots’, J. Knot Theory Ramifications 11 (2002), 1233–1242.CrossRefGoogle Scholar