Geometric learning of knot topology.

Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the conjecture that the geometrical embedding of a curve encodes information on its underlying topology is, albeit physically intuitive, far from proven.

Geometric Predictors of Knotted and Linked Arcs.

Inspired by how certain proteins "sense" knots and entanglements in DNA molecules, here, we ask if local geometric features that may be used as a readout of the underlying topology of generic polymers exist. We perform molecular simulations of knotted and linked semiflexible polymers and study four geometric measures to predict topological entanglements: local curvature, local density, local 1D writhe, and nonlocal 3D writhe. We discover that local curvature is a poor predictor of entanglements.