Developing the tube theory for polymer knots

Entanglements make polymers fundamentally different from other molecules and thus are a major theme in polymer physics research. Interchain entanglements have been extensively investigated in past decades, while intrachain entanglements, often appearing as polymer knots, are much less understood. In this work, we apply the tube theory for polymer knots based on the tube model: the polymer segments in a knot core are confined within a tube due to topological entanglements. We use an approach of visualizing and quantifying the “tubes.” First, we perform Monte Carlo simulations to generate a large number of polymer knots. Then, we superimpose knot cores to obtain average knot conformations. The fluctuations of individual knot conformations around average knot conformations produce tubes, which materialize the conceptual tubes. Analyzing the tubes validates many scaling relationships and determines the relevant parameters. Furthermore, we reveal a heart shape for polymer trefoil knots, which results from the competition of entropy and bending energy. Overall, this work builds the foundation of the tube theory for polymer knots.