Perplexing states of unentangled polymers

Polymers cannot cross, and so obey topological constraints that crucially affect their dynamics and equilibrium properties. How to deal with these non-local constraints is a major open theoretical challenge. We discuss works that provide two complementary perspectives on the field. In the first we build on topological effects to model the collapse dynamics of a polymer, as a process similar to water drops condensing on a surface. The collapsed state has attracted much attention as a model for DNA organization in the nucleus, conjectured to have a fractal structure that has so far remained elusive. Our model reproduces features of this state quantitatively, suggesting that the slow approach to scaling is related to a large dispersion in the sizes of ‘water drops’. In a second work we propose a model of unentangled directed polymers as a testing ground for theoretical ideas in the field. This model also directly relates to polymer brushes and vortex lines in superconductors. Universal quantities are found to differ significantly from predictions of the best available theories, and highlight the central role of collective many-polymer entanglement.