A ring polymer confined in a small space can be in one of two phases: expanded (dilute, dominated by solvent) or collapsed (concentrated, dominated by polymer). The transition between phases is a standard thermodynamic phenomenon — increase pressure, and the polymer collapses. The critical point, the free energy, and the osmotic pressure are determined by the polymer's interactions with itself and the confining walls. These are geometric and energetic quantities.
Janse van Rensburg, Orlandini, and Tesi (arXiv:2603.07730, March 2026) show that the thermodynamic properties of a confined ring polymer also depend on its topology — on what knot the ring polymer forms.
In a lattice model of ring polymers, they measured the osmotic pressure and free energy for different knot types: unknots, trefoil knots, and more complex non-trivial knots. Near the critical point, the free energy is measurably different for different knot types. Trefoils differ from other non-trivial knots. The topology — which crossing pattern the polymer chain forms with itself — changes the thermodynamics.
The effect is small but significant, and it is a purely topological contribution to a quantity (free energy) that is usually understood as arising from energetics and entropy. The knot constrains the conformational space available to the polymer. A trefoil knot cannot explore configurations that would require passing the chain through itself to untie. This topological restriction reduces the number of accessible microstates, which changes the entropy, which changes the free energy, which changes the osmotic pressure.
The confined setting is essential. An unconfined ring polymer has enough room that knot effects on thermodynamics are washed out by the vast conformational freedom. Confinement compresses the polymer until the topological constraints become a significant fraction of the total constraints. The knot, which was irrelevant in free space, becomes thermodynamically load-bearing when the space is small enough.
Janse van Rensburg, Orlandini, and Tesi, "Thermodynamics of Confined Knotted Lattice Polygons," arXiv:2603.07730 (March 2026).