Relative knot probabilities in confined lattice polygons

TL;DR

This work investigates how confinement within a cubic lattice cube affects knotting in ring polymers by evaluating relative knot probabilities $\rho_{n,L}(K/0_1)=p_{n,L}(K)/p_{n,L}(0_1)$ for knot types up to six crossings. Using GAS and GARM Monte Carlo sampling with BFACF moves, the authors estimate counts $p_{n,L}(K)$ and analyze how knotting depends on monomer concentration $\varphi=n/V$ and cube size $L$, defining P and $R_L$ ratios to study asymptotic behavior. Consistent with prior theory, nontrivial knots are rare relative to the unknot in small to moderate cubes, but knotting increases as $\varphi$ grows and the system nears the Hamiltonian limit; rankings show higher-crossing knots are increasingly suppressed, while certain composites and primes have distinct tendencies. The results highlight how confinement and density drive entanglement, indicating that nontrivial knots may become comparatively more common in larger, denser confining volumes, with potential universal trends awaiting confirmation through larger-scale simulations.

Abstract

In this paper we examine the relative knotting probabilities in a lattice model of ring polymers confined in a cavity. The model is of a lattice knot of size $n$ in the cubic lattice, confined to a cube of side-length $L$ and with volume $V=(L{+}1)^3$ sites. We use Monte Carlo algorithms to approximately enumerate the number of conformations of lattice knots in the confining cube. If $p_{n,L}(K)$ is the number of conformations of a lattice polygon of length $n$ and knot type $K$ in a cube of volume $L^3$, then the relative knotting probability of a lattice polygon to have knot type $K$, relative to the probability that the polygon is the unknot (the trivial knot, denoted by $0_1$), is $ρ_{n,L}(K/0_1) = p_{n,L}(K)/p_{n,L}(0_1)$. We determine $ρ_{n,L}(K/0_1)$ for various knot types $K$ up to six crossing knots. Our data show that these relative knotting probabilities are small so that the model is dominated by lattice polygons of knot type the unknot. Moreover, if the concentration of the monomers of the lattice knot is $\varphi = n/V$, then the relative knot probability increases with $\varphi$ along a curve that flattens as the Hamiltonian state is approached.

Figures (12)

• Figure 1: A schematic drawing of a knot in a confining cube or box. The side length of the cube is $L$, and the knot has rotational, translational and conformational degrees of freedom. These contribute to the free energy of the system. Notice that the cube contains $(L{+}1)^3$ lattice sites so that its volume is $V=(L{+}1)^d$ sites.
• Figure 2: A lattice knot in a cube in the cubic lattice. The side length of the cube is $L-1$ (so that it contains $V=L^3$ sites).
• Figure 3: Free energy curves of lattice unknots (see equation (\ref{['88']})) for $L\in\{6,8,10,12,14\}$. The curves appear to converge to a limiting curve with increasing $L$ and have minima at a critical concentration $\varphi_c$ (dependent on $L$). The minima are marked by a bullet on each curve.
• Figure 4: A schematic representation of the Flory-Huggins free energy. The convex curve has a minimum at $\varphi_c$, separating the model into two regimes, namely a solvent-rich (SR) phase at low concentrations $\varphi<\varphi_c$, and a polymer-rich (PR) phase at high concentrations $\varphi>\varphi_c$. The regimes are separated at the critical concentration $\varphi_c$. In the vicinity of $\varphi_c$ the system is in $\theta$-conditions.
• Figure 5: The knot probability ratio $\rho_{n,L}(3_1^+/0_1)$ of trefoils to unknots for $L=8$ (see equation (\ref{['12']})) plotted against the concentration $\varphi$. Notice the logarithmic scale and that trefoils are rare compared to unknots in this model of compressed lattice knots. With increasing concentration the ratio increases monotonically, but remains small. The confidence interval is shown by the shaded area around the curve and was determined by combining the standard deviations calculated for the raw data and then combined via the ratio in equation (\ref{['12']}).