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The statistical mechanics of a long knotted collapsed polymer is determinedby a free-energy with a knot-dependent subleading term, which is linked to thelength of the shortest polymer that can hold such knot. The only otherparameter depending on the knot kind is an amplitude such that relativeprobabilities of knots do not vary with the temperature $T$, in the limit oflong chains. We arrive at this conclusion by simulating interactingself-avoiding rings at low $T$ on the cubic lattice, both with unrestrictedtopology and with setups where the globule is divided by a slip link in twoloops (preserving their topology) which compete for the chain length, either incontact or separated by a wall as for translocation through a membrane pore.These findings suggest that in macromolecular environments there may beentropic forces with a purely topological origin, whence portions of polymersholding complex knots should tend to expand at the expense of significantlyshrinking other topologically simpler portions.

Knotted globular ring polymers: how topology affects statistics and thermodynamics