Persistent Cycle Representatives and Generalized Landscapes in Codimension 1

We develop a geometric framework for codimension 1 persistent homology by using Alexander duality to construct canonical, volume-minimizing cycle representatives in embedded filtered simplicial complexes. For a complex K, the connected components of the complement induce cycles for a homology basis in codimension 1 at each filtration value. Using the merge tree of the complement, we keep track of how these volume-optimal representatives evolve with the filtration of K, and equip each interval in the barcode with a sequence of canonical, volume-optimal representative cycles. If time permits, we also present an efficient algorithm for computing these sequences of representatives.

We apply geometric functionals to these representative cycles, such as path length, enclosed volume, or excess curvature. This way, we obtain a real-valued function for each interval, which captures geometric information about K. Deriving from this construction, we introduce generalized persistence landscapes, which specialize to standard persistence landscape with the constant-one functional. Generalized landscapes can distinguish point clouds with similar persistent homology but distinct shape, which we demonstrate by concrete examples.