Take a closed loop embedded in three-dimensional space --- a knot, or a simple circle, or anything in between. For every pair of points on that loop, measure the Euclidean distance between them. This produces a function defined not on the loop itself but on the space of all unordered pairs of points on it --- a space that is topologically a Mobius band. The resulting function, called the chordal distance transform, encodes the shape of the embedding in a way that is invariant to both rigid motions and reparametrizations of the curve.
The sublevel sets of this function --- the regions where all chords are shorter than some threshold --- evolve as the threshold increases. Their persistent homology, which tracks the birth and death of topological features across this filtration, produces a barcode that serves as a shape descriptor. The descriptor is continuous: small perturbations of the embedding produce small changes in the barcode, measured in bottleneck distance. It is also generic: for smooth embeddings, the critical points of the chordal distance transform are finite and non-degenerate on an open dense set, ensuring the barcode is well-behaved for typical curves.
The geometric meaning of these critical points is concrete. A critical point of the chordal distance transform corresponds to a chord that is perpendicular to the curve at both endpoints --- a locally extremal connection between two parts of the loop. These special chords are the structural skeleton of the embedding, the places where the curve's relationship to itself changes character.
The through-claim is that the right way to characterize a shape is often not to study the shape directly but to study the distances within it. The function that measures internal separation can be more informative than the object it measures. The map of distances is richer than the territory.