A statistical approach to knot confinement via persistent homology

Abstract

In this paper, we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot’s embedding for growing radii. Specifically, we extract features from the persistent homology (PH) of the Vietoris–Rips complexes built from point clouds associated with knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated with the embedding and PH-based features, as a function of the knots’ lengths. We further study the variation of these correlations for different knot types. Finally, this framework also allows us to define a simple notion of deviation from ideal configurations of knots.

Homology of homologous knotted proteins

“…In this work, we combine PH and low-dimensional topology to characterize geometric features of (open) knotted proteins. To our knowledge, the only other work linking PH and low-dimensional topology is new and considers only closed knots [ 32 ]. In a different direction, Mapper [ 33 ], another tool from computational topology, has been applied to the abstract collection of closed knots [ 34 ] parametrized by the values taken by polynomial invariants rather than the geometry of any specific embeddings.…”