The free energy of a model of uniformly weighted lattice knots of length $n$ and knot type $K$ confined to a lattice cube of side length $L-1$ is given by $F_L\left(\varphi \right)=-\frac{1}{V}log{p}_{n,L}\left(K\right)$, where $V=L^3$ and where $\varphi =n/V$ is the concentration of monomers of the lattice knot in the confining cube. The limiting free energy of the model is $F_{\infty }\left(\varphi \right)={lim}_{L\to \infty }{F}_L\left(\varphi \right)$ and the limiting osmotic pressure of monomers leaving the lattice knot to become solvent molecules is defined by ${\mathrm{\Pi }}_{\infty }\left(\varphi \right)={\varphi }^2\frac{d}{d\varphi }[F_{\infty }\left(\varphi \right)/\varphi ]$. I show that, under very mild assumptions, the functions $P_L\left(\varphi \right)={\varphi }^2\frac{d}{d\varphi }[F_L\left(\varphi \right)/\varphi ]{|}_n$ and ${\mathrm{\Pi }}_L\left(\varphi \right)={\varphi }^2\frac{d}{d\varphi }[F_L\left(\varphi \right)/\varphi ]{|}_L$ are finite-size approximations of ${\mathrm{\Pi }}_{\infty }\left(\varphi \right)$.

• Received 2 December 2019

DOI:https://doi.org/10.1103/PhysRevE.101.016502