Knotting in a lattice model of confined ring polymers

Polymers in confined spaces or in dense melts are known to have increased entanglements and self-entanglements.  These entanglements may manifest as an increased probability of local and global knotted segments along the polymer.  In this talk we examine knotting in a lattice model of a confined ring polymer.   In particular, we examining relative knotting probabilities in a lattice ring polymer (a polygon) of length n in the cubic lattice, confined and compressed in a cube of side-length L.  This is a model of a confined and compressed lattice knot.  Confined lattice knots of length n in a cube of side-length L can be sampled using Monte Carlo algorithms.   The GAS and GARM algorithms was used to estimate the number of its conformations p_n,L(K) of a confined lattice knot of length n, in a cube of side-length L and of fixed knot type K. If K is the unknot (also denoted by 0₁) then the lattice knot is said to have knot type the unknot (the trivial knot).  The relative knotting probability of a confined lattice knot of (non-trivial) knot type K with respect to the unknot is given by 

ρ_n,L(K/0₁) = p_n,L(K)/p_n,L(0₁) .

ρ_n,L(K/0₁) was estimated for various knot types K up to six crossing knots. The data show that ρ_n,L(K/0₁) is small over a wide range of the concentration φ of monomers for cubes of side-lengths up to L=12.  Generally the unknot is the dominating knot type in this model, but ρ_n,L(K/0₁) increases with concentration φ along a curve that flattens as the Hamiltonian state is approached.  These results strongly suggest that as the Hamiltonian state is approached, the lattice knot delocalizes or "melts" into the dense and entangled confined polymer.  This is in contrast  with lattice knot models of (unconfined) knotted ring polymers, where knotted arcs along the polymer backbone are confined or "localized" in small balls along the backbone [Orlandini etal, J Phys A: Math Gen Vol 31 p5953(1998)].  This work was done in collaboration with MC Tesi and E Orlandini and publised in Physical Review E 111 065406 (2025).