Relative knot probabilities in confined lattice polygons

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 estimate approximately the number of conformations of lattice knots in the confining cube. If pn,L(K) is the number of conformations of a lattice polygon of length n and knot type K in a cube of volume L3, 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 01), is ρn,L(K/01)=pn,L(K)/pn,L(01). We determine ρn,L(K/01) for various knot types K up to six crossing knots. Our data show that these relative knotting probabilities are small over a wide range of the concentration φ=n/V of monomers for values of L≤12 so that the model is dominated by unknotted lattice polygons. Moreover, the relative knot probability increases with φ along a curve that flattens as the Hamiltonian state is approached.