Opacity and Entanglement of Polymer Chains

We argue that the mean crossing number of a random polymer configuration is simply a measure of opacity, without being closely related to entanglement as claimed by several authors. We present an easy way of estimating its asymptotic behaviour numerically. These estimates agree for random walks (theta polymers), self-avoiding walks, and for compact globules with analytic estimates giving $\log N,a-b\mathrm{/}{N}^{2\nu -1},$ and $N^{1\mathrm{/}3}$, respectively, for the average number of crossings per monomer in the limit $N\to \mathrm{\infty }$. While the result for compact globules agrees with a rigorous previous estimate, the result for SAWs disagrees with previous numerical estimates.