Knots, entropy

The connection between problem of determination of the knot entropy and statistics of entangled random walks is schematically shown in the Table 1. We argue that both these topological questions could be considered from one general point of view of random walk on non commutative groups. (Some preliminary remarks concerning this connection one can find in [2].)... 

Thus, it should be stressed that the mathematical topological theory investigates, as a rule, the problems of classification of knots and links, the construction of topological invariants, definitions of topological classes, etc. whereas the fundamental physical problem in the theory of topological properties of polymer chains is the determination of the entropy, S = In Z with the fixed topological state of chains. Both these problems are very difficult, but important. 

Our main aim concerns the determination of the entropy of the knot embedded in 3D space (or, in other words, the determination of available volume in the phase space for the path with fixed topological state). 

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