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This paper introduces a method to detect each geometrically significant loopthat is a geodesic circle (an isometric embedding of $S^1$) and a bottleneckloop (meaning that each of its perturbations increases the length) in ageodesic space using persistent homology. Under fairly mild conditions we showthat such a loop either terminates a $1$-dimensional homology class or givesrise to a $2$-dimensional homology class in persistent homology. The main toolin this detection technique are selective Rips complexes, new custom madecomplexes that function as an appropriate combinatorial lens for persistenthomology in order to detect the above mentioned loops. The main argument isbased on a new concept of a local winding number, which turns out to be aninvariant of certain homology classes.

Persistent Homology with Selective Rips complexes detects geodesic circles