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

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

Candidate Tidal Dwarf Galaxies in the Compact Group CG J1720-67.8

This is the second part of a detailed study of the ultracompact group CG J1720-67.8: in the first part we have focused the attention on the three main galaxies of the group and we have identified a number of candidate tidal dwarf galaxies (TDGs). Here we concentrate on these candidate TDGs. Absolute photometry of these objects in BVRJHKs bands confirms their relatively blue colors, as we already expected from the inspection of optical and near-infrared color maps and from the presence of emission-lines in the optical spectra. The physical conditions in such candidate TDGs are investigated through the application of photoionization models, while the optical colors are compared with grids of spectrophotometric evolutionary synthesis models from the literature. Although from our data self-gravitation cannot be proved for these objects, their general properties are consistent with those of other TDG candidates. Additionally we present the photometry of a few ``knots'' detected in the immediate surroundings of CG J1720-67.8 and consider the possibility that these objects might belong to a dwarf population associated with the compact group.Comment: Accepted for publication in the Astrophysical Journa