Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Crosscap Numbers of Two-component Links

  • Zhang, Gengyu (Department of Mathematics, Tokyo Institute of Technology)
  • Received : 2006.06.29
  • Published : 2008.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

We define the crosscap number of a 2-component link as the minimum of the first Betti numbers of connected, non-orientable surfaces bounding the link. We discuss some properties of the crosscap numbers of 2-component links.

Keywords

References

  1. B. E. Clark, Crosscaps and knots, Internat. J. Math. Sci., 1(1)(1978), 113-223, MR 0478131 (57 #17620). https://doi.org/10.1155/S0161171278000149
  2. C. McA. Gordon and R. A. Litherland, On the signature of a link, Invent. Math., 47(1)(1978), 53 - 69, MR 0500905 (58 #18407). https://doi.org/10.1007/BF01609479
  3. M. Hirasawa and M. Teragaito, Crosscap numbers of 2-bridge knots, Topology, 45(3)(2006), 513 - 530. https://doi.org/10.1016/j.top.2005.11.001
  4. K. Ichihara and S. Mizushima, Crosscap numbers of pretzel knots, arXiv : math.GT/0608480.
  5. W. B. R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997, MR 1472978 (98f:57015).
  6. H. Murakami and A. Yasuhara, Crosscap number of a knot, Pacific J. Math., 171(1)(1995), 261 - 273, MR 1362987 (96k:57006). https://doi.org/10.2140/pjm.1995.171.261
  7. K. Murasugi, On the signature of links, Topology, 9(1970), 283 - 298, MR 0261585 (41 #6198). https://doi.org/10.1016/0040-9383(70)90018-2
  8. I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley & Sons Inc., New York, 1991, MR 1083765 (91i:11001).
  9. D. Rolfsen, Knots and Links, Mathematics Lecture Series, vol. 7, Publish or Perish Inc., Houston, TX, 1990, MR 95c:57018.
  10. M. Teragaito, Crosscap numbers of torus knots, Topology Appl., 138(1-3)(2004), 219- 238, MR 2035482 (2005a:57007). https://doi.org/10.1016/j.topol.2003.08.004
  11. H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math., 76(2)(1962), 464 - 498, MR 0143201 (26 #761). https://doi.org/10.2307/1970369

Cited by

  1. Symmetrically Bordered Surfaces vol.117, pp.7, 2010, https://doi.org/10.4169/000298910x496705
  2. A classification of spanning surfaces for alternating links vol.13, pp.5, 2013, https://doi.org/10.2140/agt.2013.13.2967