Morse frames
This work addresses computational topology problems for researchers in discrete Morse theory and homology, offering incremental improvements to existing methods.
The paper introduces Morse frames, which map critical simplexes to all simplexes in discrete Morse theory, with Morse references as the main example enabling computation of Morse complexes for homology analysis. It presents a novel variant of the Annotation algorithm for persistent cohomology using Morse frames and a construction leveraging Morse references to compute Betti numbers in mod 2 arithmetic.
In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references allow computing Morse complexes, an important tool for homology. We highlight the link between Morse references and gradient flows. We also propose a novel presentation of the Annotation algorithm for persistent cohomology, as a variant of a Morse frame. Finally, we propose another construction, that takes advantage of the Morse reference for computing the Betti numbers in mod 2 arithmetic.