  Understanding the biological function of knots in proteins and their folding
process is an open and challenging question in biology. Recent studies classify
the topology and geometry of knotted proteins by analysing the distribution of
a protein's planar projections using topological objects called knotoids. We
approach the analysis of proteins with the same topology by introducing a
topologically inspired statistical metric between their knotoid distributions.
We detect geometric differences between trefoil proteins by characterising
their entanglement and we recover a clustering by sequence similarity. By
looking directly at the geometry and topology of their native states, we are
able to probe different folding pathways for proteins forming open-ended
trefoil knots. Interestingly, our pipeline reveals that the folding pathway of
shallow knotted Carbonic Anhydrases involves the creation of a double-looped
structure, differently from what was previously observed for deeply knotted
trefoil proteins. We validate this with Molecular Dynamics simulations.
