  The statistical mechanics of a long knotted collapsed polymer is determined
by a free-energy with a knot-dependent subleading term, which is linked to the
length of the shortest polymer that can hold such knot. The only other
parameter depending on the knot kind is an amplitude such that relative
probabilities of knots do not vary with the temperature $T$, in the limit of
long chains. We arrive at this conclusion by simulating interacting
self-avoiding rings at low $T$ on the cubic lattice, both with unrestricted
topology and with setups where the globule is divided by a slip link in two
loops (preserving their topology) which compete for the chain length, either in
contact or separated by a wall as for translocation through a membrane pore.
These findings suggest that in macromolecular environments there may be
entropic forces with a purely topological origin, whence portions of polymers
holding complex knots should tend to expand at the expense of significantly
shrinking other topologically simpler portions.
