The role of curvature in entanglement

We offer a resolution to this conundrum. We show, that at least for simple but generally applicable models, the relationship between curvature and entanglement is subtle: if we keep filament density constant and increase curvature, the entanglement initially increases, passes through a maximum, then decreases, so there is a regime where increasing curvature increases entanglement, and there is also a regime where increasing curvature decreases entanglement. This has implications for filament packing in many circumstances, and in particular for the compaction structure of DNA in the cell-it provides a straightforward argument for the view that one purpose of DNA coiling and supercoiling is to inhibit entanglement. It also tells us to expect that wavy hair-neither the straightest nor the curliest-tangles most readily.