ON TOPOLOGICAL CORRELATIONS IN TRIVIAL KNOTS: FROM BROWNIAN BRIDGES TO CRUMPLED GLOBULES

Abstract

We study, analytically and by numerical simulations, a mathematical model of closed unknotted polymer chains. The polymers are modeled by what we call "densely packed knots", which are knot diagrams entirely covering a long rectangular lattice, with randomly generated overpasses and underpasses at the nodes of the lattice. We are interested in the probability of the "knottedness" (the "knot complexity") of densily packed knots, which we measure via the degree of their Jones–Kauffman polynomial. In particular, we find the mean complexity of "daughter knots", obtained by cutting off a part of a trivial (i.e. unknotted) "parent" densily packed knot and closig up the "open tails" (the loose ends of the cut strands). We present arguments supporting the conjecture that the knot complexity n* of a daughter knot of an unknotted parent one grows as [Formula: see text] where N is the total number of vertices on the lattice. This result gives a strong support for the conjectured "crumpled globule" structure of collapsed unknotted closed polymer chains, in which the polymer forms a system of densely packed folds, mutually separated in a broad range of scales.