Theoretical and Mathematical Physics

, Volume 120, Issue 2, pp 985–996 | Cite as

Upper estimate of the cardinality of the set of knots generated by one-and two-dimensional braids

  • R. R. Bikbov
  • S. K. Nechaev


We give the upper estimate for the cardinality of the set Ω(n, μ) of knots generated by closed one- and two-dimensional braids with n generators of the irreducible length μ in the limit as n≫1 and μ≫1.


Free Group Homotopy Class Braid Group Normal Order Cayley Tree 
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  1. 1.
    J. Nadal, B. Derrida, and J. Vannimenus,J. Physique,43, 1561 (1982); V. Hakim and J. Nadal,J. Phys. A,16, L213 (1983).MathSciNetGoogle Scholar
  2. 2.
    A. Vershik, S. Nechaev, and R. Bikbov,Commun. Math. Phys. (forthcoming).Google Scholar
  3. 3.
    L. Paris and D. Rolfsen, “Geometric subgroups of surface braid groups,” Preprint 115, Université de Bourgogne, Laboratoire de Topologie, Dijon (1997).Google Scholar
  4. 4.
    J. Birman,Contemp. Math.,78, 13 (1988).MathSciNetGoogle Scholar
  5. 5.
    A. M. Vershik and S. V. Kerov,Sov. Math. Dokl.,38, No. 1, 134 (1989); A. M. Vershik,Topics Alg.,26, 467 (1990);Proc. Am. Math. Soc.,148, 1 (1991).zbMATHMathSciNetGoogle Scholar
  6. 6.
    S. K. Nechaev, A. Yu. Grosberg, and A. M. Vershik,J. Phys. A,29, 2411 (1996).zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    P. Cartier and D. Foata,Problémes combinatoires de commutation et réarrangement (Lect. Notes Math., Vol. 85), Springer, New York (1969).Google Scholar
  8. 8.
    G. X. Viennot, “Heaps of pieces I: Basic definitions and combinatorial lemmas,” in:Combinatoire énumérative. Proc. “Colloque de combinatoire énumérative,” Université du Québec à Montréal, May 28-June 1, 1985 (G. Labelle et P. Leroux, eds.) (Lect. Notes Math., Vol. 1234), Springer, Berlin (1986), p. 321.Google Scholar
  9. 9.
    A. M. Vershik,Usp. Mat. Nauk, No. 5, 57 (1999).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • R. R. Bikbov
    • 1
  • S. K. Nechaev
    • 1
    • 2
  1. 1.Landau Institute for Theoretical physicsRussian Academy of SciencesMoscowRussia
  2. 2.LPTMSUniversité Paris SudOrsay CedexFrance

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