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Thermodynamics and topology of disordered systems: Statistics of the random knot diagrams on finite lattices

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Abstract

The statistical properties of random lattice knots, the topology of which is determined by the algebraic topological Jones-Kauffman invariants, was studied by analytical and numerical methods. The Kauffman polynomial invariant of a random knot diagram was represented by a partition function of the Potts model with a random configuration of ferro-and antiferromagnetic bonds, which allowed the probability distribution of the random dense knots on a flat square lattice over topological classes to be studied. A topological class is characterized by the highest power of the Kauffman polynomial invariant and interpreted as the free energy of a q-component Potts spin system for q→∞. It is shown that the highest power of the Kauffman invariant correlates with the minimum energy of the corresponding Potts spin system. The probability of the lattice knot distribution over topological classes was studied by the method of transfer matrices, depending on the type of local junctions and the size of the flat knot diagram. The results obtained are compared to the probability distribution of the minimum energy of a Potts system with random ferro-and antiferromagnetic bonds.

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Authors and Affiliations

  1. Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432, Russia

    O. A. Vasilyev & S. K. Nechaev

  2. UMR 8626, CNRS-Université Paris XI, LPTMS, Bat. 100, Université Paris Sud, 91405, Orsay Cedex, France

    S. K. Nechaev

Authors
  1. O. A. Vasilyev
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  2. S. K. Nechaev
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Translated from Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) Fiziki, Vol. 120, No. 5, 2001, pp. 1288–1308.

Original Russian Text Copyright © 2001 by Vasilyev, Nechaev.

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Vasilyev, O.A., Nechaev, S.K. Thermodynamics and topology of disordered systems: Statistics of the random knot diagrams on finite lattices. J. Exp. Theor. Phys. 93, 1119–1136 (2001). https://doi.org/10.1134/1.1427184

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  • DOI: https://doi.org/10.1134/1.1427184

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