Applications of Knot Theory to Characterizing Entanglements for Polymers in Tubes or Nano-channels

May 23, 2023
Motivated in part by recent experimental data for knotting of DNA in nano-channels and nano-pores, we have been studying the entanglement complexity of polymers confined to lattice tubes. To model ring polymers in solution, we consider systems of polygons confined to tubular subsets of the simple cubic lattice and study their entanglement complexity using a combination of knot theory, transfer-matrix methods and Monte Carlo computer simulation. For the smallest tube that admits knotting, we prove long-standing conjectures about the knot and link statistics as the system size grows. Monte Carlo simulations are used to explore the conjectures for larger tube sizes. I will review these results including our most recent results for the special case of a system of two polygons that span the tube.