Analytical estimation of scaling behavior for the entanglement complexity of a bond network

Geometrical entanglements in a polymer network can be characterized in terms of the mean number of projected bond-bond crossings, N. Here, we present an analytical method to study the dependence of N on the number of bonds in the network, n. Our approach shows the occurrence of power-law scaling, N - n(beta). The estimated upper bound to the exponent for maximally compact networks, Beta approximately 1.38, agrees well with the values observed in simulations of transient networks in liquids and in the folding features of native states of globular proteins.