The Seifert Matrices of Periodic Links with Rational Quotients

  • Lee, Sang Youl (Department of Mathematics, Pusan National University) ;
  • Park, Maeng-Sang (Department of Mathematics, Graduate School of Natural Sciences, Pusan National University) ;
  • Seo, Myoungsoo (Department of Mathematics, Pusan National University)
  • Received : 2006.10.23
  • Published : 2007.06.23

Abstract

In this paper, we characterize the Seifert matrices of $p$-periodic links whose quotients are 2-bridge links $C(2,n_1,-2,n_2,{\cdots},n_r,(-2)^r)$ and give formulas for the signatures and determinants of the 3-periodic links of these kind in terms of $n_1$, $n_2$, ${\cdots}$, $n_r$.

Keywords

References

  1. J. H. Conway, An enumeration of knots and links and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf. Oxford) (J. Leech, ed.), Pergamon Press, New York, 1970, 329-358.
  2. C. McA. Gordon and R. A. Litherland, On the signature of a link, Invent. Math., 47(1978), 53-69. https://doi.org/10.1007/BF01609479
  3. C. McA. Gordon and R. A. Litherland , On the theorem of Murasugi, Pacific J. Math., 82(1979), 69-74. https://doi.org/10.2140/pjm.1979.82.69
  4. C. McA. Gordon and R. A. Litherland and K. Murasugi, Signatures of covering links, Canad. J. Math., 33(1981), 381-394. https://doi.org/10.4153/CJM-1981-032-3
  5. H. M. Hilden, M. T. Lozano, and J. M. Montesinos-Amilibia, On the character variety of periodic knots and links, Math. Proc. Cambridge Philos. Soc., 129(2000), 477-490. https://doi.org/10.1017/S0305004100004679
  6. T. Kanenobu and Y. Miyazawa, 2-bridge link projections, Kobe J. Math., 9(1992), 171-182.
  7. H. J. Jang, S. Y. Lee and M. Seo, Casson knot invariants of periodic knots with rational quotients, J. Knot Theory Ramifications, 16(2007), 439-460. https://doi.org/10.1142/S0218216507005373
  8. S. Y. Lee, $Z_{n}$-equivalant Goeritz matrices of periodic links, Osaka J. Math., 40(2003), 393-408.
  9. S. Y. Lee and C-Y. Park, On the modified Goeritz matrices of 2-periodic links, Osaka J. Math., 35(1998), 529-537.
  10. S. Y. Lee and M. Seo, Recurrence formulas for the Alexander polynomials of 2-bridge links and their covering links, J. Knot Theory Ramifications, 15(2006), 179-203. https://doi.org/10.1142/S0218216506004403
  11. S. Y. Lee and M. Seo, Casson knot invariants of periodic knot with rational quotients II, J. Knot Theory Ramifications(to appear).
  12. K. Murasugi, On the signature of links, Topology, 9(1970), 283-298. https://doi.org/10.1016/0040-9383(70)90018-2
  13. K. Murasugi, Knot theory and its application, Birkhauser, 1996.
  14. H. Schubert, Knoten mit zwei Brucken, Math Z., 65(1956), 133-170. https://doi.org/10.1007/BF01473875
  15. H. Seifert, Uber das Geschlecht von Knoten, Math. Ann., 110(1934) 571-592.