Path-integral calculation of the mean number of overcrossings in an entangled polymer network

A quantitative characterization of entanglements between polymer chains is essential for understanding the behavior of polymer solutions. When dealing with unknotted open chains, that characterization must rely on geometrical (rather than topological) measures of entanglement complexity. In this work, we deal with a simple geometrical shape descriptor: the mean overcrossing number of a set of curves. This descriptor provides a physically intuitive characterization of self-entanglements in a chain or entanglements in a network. Many of the analytical properties of the mean overcrossing number are still not well understood. In part, this is due to the use of numerical algorithms for its computation. Path-integral formalisms offer an improvement over this situation, by providing analytical expressions for the geometrical descriptors or, at least, more efficient algorithms for their evaluation. In this work, we discuss an approach to represent: overcrossing numbers by path integrals. The formalism is general enough to be applied to polymer networks. In the particular case of networks on the cubic lattice, we provide a set of closed formulas for the fast (exact) calculation of the mean overcrossing number. By using our methodology, it is now possible to perform efficient analyses of entanglement complexity in polymer solutions modeled by computer simulations on lattices. Our results can be used to gain a better understanding of the dynamical behavior of entangled polymers.