Counting polymer knots to find the entanglement length

Uncrossability of polymer chains in a melt gives rise to a restricted transverse motion of chains, which is represented by a confining "tube". Ultimately, the tube must be of topological origin. We propose two definitions of the tube diameter or entanglement length (N _e) in terms of the properties of topologically equilibrated melts of rings: (1) the probability of a ring in such a melt being unknotted is a constant for a ring of length N _e; (2) the topological entropy per entanglement strand is 3/2k _B for sufficiently long rings. To test these ideas, we simulated a coarse-grained model for polymer rings under aperiodic, 1D and 2D periodic boundary conditions, with molecular rebridging moves to equilibrate the topological states. We then implemented an efficient algorithm for computing the Jones polynomial, in order to identify the topological states. Our purely topological estimates of N _e are quite consistent with previous values based on heuristic chain-shrinking methods.