DOI QR코드

DOI QR Code

DEALTERNATING NUMBERS AND CLASSICAL LINK INVARIANTS

  • Received : 2011.03.07
  • Accepted : 2011.11.14
  • Published : 2012.01.31

Abstract

Dealternating numbers and alternation numbers measure the distance between the link and an alternating links. In the present article, we show that classical link invariants; the determinant, signature and Alexander polynomial can not detect the almost alternativity of links.

Keywords

References

  1. T. Abe, An estimation of the alternation number of a torus knot, J. Knot Theory Ramifications 18(3) (2009), 363-379. https://doi.org/10.1142/S021821650900694X
  2. C. Adams, Almost alternating links, Topology and its applications 46 (1992), 151-165. https://doi.org/10.1016/0166-8641(92)90130-R
  3. C. Adams, The knot book, W. H. Freeman and Company. 1994.
  4. T. Abe and K. Kishimoto, The dealternation number and the alternation number of a closed 3-braid, J. Knot Theory Ramifications 19(9) (2010), 1157-1181. https://doi.org/10.1142/S0218216510008352
  5. J. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational problemas in abstract algebra, Pergamon Press, 1969.
  6. R. Crowell, Genus of alternating link types, Ann. of Math. 69 (1959), 258-275. https://doi.org/10.2307/1970181
  7. R. H. Fox, Some problems in knot theory, Prentice-Hall, 1962.
  8. C. McA. Gordon and R. Litherland, On the signature of a link, Invent. Math. 47 (1987), 53-69.
  9. J. Hoste, M. Thistlethwaite and J. Weeks, The First 1701936 Knots, Math. Intell. 20 (1998), 33-48. https://doi.org/10.1007/BF03025227
  10. T. Kanenobu, Upper bound for the alternation number of a torus knot, Topology Appl., in Press.
  11. A. Kawauchi, On alternation numbers of links, Topology Appl., in Press.
  12. D. Kim and J. Lee, On pretzel links, Bull. of Aust. Math. Soc. 75(2) (2007), 253-271. https://doi.org/10.1017/S0004972700039198
  13. W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), 37-44. https://doi.org/10.1016/0040-9383(84)90023-5
  14. K. Murasugi, On the Alexander polynomial of alternating algebraic knots, J. of Aus. Math. Soc. 39 (1985), 317-333. https://doi.org/10.1017/S1446788700026094
  15. T. Van Zandt. PSTricks: PostScript macros for generic TEX. Available at ftp://ftp.princeton.edu/pub/tvz/.