  The lectures review the state of affairs in modern branch of mathematical
physics called probabilistic topology. In particular we consider the following
problems: (i) We estimate the probability of a trivial knot formation on the
lattice using the Kauffman algebraic invariants and show the connection of this
problem with the thermodynamic properties of 2D disordered Potts model; (ii) We
investigate the limit behavior of random walks in multi-connected spaces and on
non-commutative groups related to the knot theory. We discuss the application
of the above mentioned problems in statistical physics of polymer chains. On
the basis of non-commutative probability theory we derive some new results in
statistical physics of entangled polymer chains which unite rigorous
mathematical facts with more intuitive physical arguments.
