Knot probabilities in a lattice knot model of compressed ring polymers

The entropy of knotted ring polymers can be quantified by lattice knots. This model gives qualitative insights into the knotting probabilities and entanglement complexity of ring polymers. The Frisch-Wasserman-Delbruck conjecture that the knotting probability of a ring polymer approaches one was verified for lattice knots in 1988 (J Phys A: Math Gen 21 p1689 (1988) DW Sumners & SG Whittington) while numerical work shows that knotting probability in models of lattice knots increases slowly with increasing length (J Phys A: Math Gen 23 p 3573 (1990) EJ Janse van Rensburg & SG Whittington). However, little is known about entanglements and knotting probabilities of ring polymers in confining cavities or pores, and this will be explored in this talk, where I shall review a lattice knot model of confined ring polymers. Numerical data for this model will be presented, extending older results on the free energies of confined lattice knots (Phys Rev E 100 p012501 (2019) & J Phys A: Math Theo 53 p015002 (2019) EJ Janse van Rensburg) and square lattice links (Phys Rev E 104 p064134 (2021) EJ Janse van Rensburg & E Orlandini). I shall briefly review what is known about knotting in lattice polygons, and how to approximately enumerate lattice knots of fixed knot types. Data on the relative knotting probabilities of confined lattice polygons will be presented, also into the dense compressed regime of the model. These results show that the unknot is the most likely knot type, even in a highly compressed environment (approaching the Hamiltonian circuits of a cube). Additional results on the relative probabilities of non-trivial knot types will be presented as well. In particular, increasing knot complexity (as measured by the minimal crossing number in a knot projection) suppresses the relative knot probability. However, knot types with the same minimal crossing number (such as the 5 crossing knots 5_1 and 5_2) may have different relative knotting probabilities. A manuscript based on these results is in preparation.