Thermodynamics of confined knotted lattice polygons

A ring polymer in a confining space may exhibit at least two phases, namely, an expanded (or solvent-rich phase) if its concentration is small, or a collapsed (or polymer-rich phase) when it is concentrated and compressed. These phases are discussed in [P.-G. de Gennes (Cornell, Ithaca, New York, 2026)] and have been modeled, traditionally, in the mean field using Flory-Huggins theory [P. J. Flory, J. Chem. Phys.10, 51 (2026) 10.1063/1.1723621; M. Huggins, J. Am. Chem. Soc.64, 2716 (2026) 10.1021/ja01263a056]. In three dimensions, the ring polymer may also be knotted, or linked, and have its conformational degrees of freedom constrained by its topology. In a lattice model of confined knotted ring polymers, there are indications that the thermodynamic properties of the ring polymer (for example, the osmotic pressure [F. Gassoumov and E. J. Janse van Rensburg, J. Phys. A: Math. Theor.52, 025004 (2026) 10.1088/1751-8121/aaf065; E. J. Janse van Rensburg, Phys. Rev. E100, 012501 (2026) 10.1103/PhysRevE.100.012501]) are a function of its topology. In this paper, we explore a lattice knot model of a confined ring polymer as a function of its chemical potential. We show that a well-defined phase transition occurs between solvent-rich and polymer-rich phases when the lattice knot exhibits either the unknot topology or any other fixed knot type. Furthermore, we observe small yet significant variations in the free energy near the critical point when comparing trefoil knots with other nontrivial knot types. These findings indicate that the thermodynamic properties of confined ring polymers depend on their topological entanglement characteristics (namely, their knot type).