Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the Lickorish-Millett-Turaev formula for the Kauffman polynomial


Author: Józef H. Przytycki
Journal: Proc. Amer. Math. Soc. 121 (1994), 645-647
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9939-1994-1213869-5
MathSciNet review: 1213869
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use the idea of expressing a nonoriented link as a sum of all oriented links corresponding to the link to present a short proof of the Lickorish-Millett-Turaev formula for the Kauffman polynomial at $ z = - a - {a^{ - 1}}$. Our approach explains the observation made by Lickorish and Millett that the formula is the generating function for the linking number of a sublink of the given link with its complementary sublink.


References [Enhancements On Off] (What's this?)

  • [1] W. Goldman, Invariant functions of Lie groups and Hamiltonian flows of surface group representation, Invent. Math. 85 (1986), 263-302. MR 846929 (87j:32069)
  • [2] J. Hoste and J. H. Przytycki, Homotopy skein modules of orientable 3-manifolds, Math. Proc. Cambridge Philos. Soc. 108 (1990), 475-488. MR 1068450 (91g:57006)
  • [3] -, A survey of skein modules of 3-manifolds, "Knots 90", De Gruyter, Berlin, New York, 1992, pp. 363-379. MR 1177433 (93m:57018)
  • [4] L. H. Kauffman, On knots, Ann. of Math. Stud., vol. 115, Princeton Univ. Press, Princeton, NJ, 1987.
  • [5] W. B. R. Lickorish and K. C. Millett, An evaluation of the F-polynomial of a link, Differential Topology, Proc. 2nd Topology Symp. (Siegen/FRG 1987), Lecture Notes in Math., vol. 1350, Springer-Verlag, New York, 1988, pp. 104-108. MR 979335 (89m:57008)
  • [6] A. S. Lipson, Some more states models for link invariants, Pacific J. Math. 152 (1992), 337-346. MR 1141800 (92i:57004)
  • [7] H. R. Morton, Problems, Braids (J. S. Birman and A. Libgober, eds.), Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 557-574. MR 975094
  • [8] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527-553. MR 939474 (89e:57003)
  • [9] -, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4) 4 (1991), 635-704. MR 1142906 (94a:57023)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57M25

Retrieve articles in all journals with MSC: 57M25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1213869-5
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society