  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.
