Statistics of Knots and Entangled Random Walks

In this book, the author announces the class of problems called "entropy of knots" and gives the overview of existing topological invariants. He constructs statistical models on braids using the representations of Alexander and Jones invariants and puts forward the question of limit distribution of these invariants for randomly generated braids. The relation of highest powers of corresponding algebraic invariants to the Lyapunov exponents of the products of noncommunicative matrices is shown. Also the problem of conditional joint limit distribution for "brownian bridges" on braids is discussed. Special cases of noncommutative groups PSL(2,R), PSL(2,Z) and braid groups are considered in detail. In the volume, the author also discusses the application of conformal methods for the explicit construction of topological invariants for random walks on multiconnected manifolds. Furthermore the connection of these topological invariants and the monodromy properties of correlation functions of some conformal theories are also discussed.