Thermodynamics and Topology of Disordered Systems: Statistics of the Random Knot Diagrams on Finite Lattice

Details Thermodynamics and Topology of Disordered Systems: Statistics of the Random Knot Diagrams on Finite Lattice Thermodynamics and Topology of Disordered Systems: Statistics of the Random Knot Diagrams on Finite Lattice

2001-11-06

arxiv.org/abs/cond-mat/0111091v2

JETP, 93 (2001), 1119--1136

Physics

Condensed Matter

Disordered Systems and Neural Networks

37 pages, latex-revtex (new version: misprints removed, references added)

Scientific paper

10.1134/1.1427184

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->infinity. It is shown that the highest power of the Kauffman invariant is correlated 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 obtained results are compared to the probability distribution of the minimum energy of a Potts system with random ferro- and antiferromagnetic bonds.

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