  Quantification and classification of protein structures, such as knotted
proteins, often requires noise-free and complete data. Here we develop a
mathematical pipeline that systematically analyzes protein structures. We
showcase this geometric framework on proteins forming open-ended trefoil knots,
and we demonstrate that the mathematical tool, persistent homology, faithfully
represents their structural homology. This topological pipeline identifies
important geometric features of protein entanglement and clusters the space of
trefoil proteins according to their depth. Persistence landscapes quantify the
topological difference between a family of knotted and unknotted proteins in
the same structural homology class. This difference is localized and
interpreted geometrically with recent advancements in systematic computation of
homology generators. The topological and geometric quantification we find is
robust to noisy input data, which demonstrates the potential of this approach
in contexts where standard knot theoretic tools fail.
