  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 of the
Vietoris-Rips complexes built from point clouds associated to knots.
Statistical analysis of our data shows the existence of increasing correlations
between geometric quantities associated to the embedding and persistent
homology 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.
