TY - JOUR TI - Computational connection matrix theory DO - https://doi.org/doi:10.7282/t3-kh58-md25 PY - 2019 AB - We develop a computational and categorical framework for connection matrix theory. In terms of computations, we give an algorithm for computing the connection matrix based on algebraic-discrete Morse theory. The makes the connection matrix available, for the first time, as a computational tool within applied topology and dynamics. In addition, the algorithm provides a straightforward constructive proof of the existence of connection matrices. In terms of categories, our formulation resolves the non-uniqueness of the connection matrix, as well as relates the connection matrix to persistent homology. We extend existing computational Conley theory to incorporate connection matrix theory. This is done by developing a setting, which we call transversality models, in which discrete approximations to continuous flows can be used to compute connection matrices for the underlying continuous system. We make applications to a Morse theory on spaces of braid diagrams. Finally, we provide an implicit discrete Morse pairing for cubical complexes. This enables the computation of connection matrices in high-dimensional cubical complexes. We benchmark our algorithm on a set of such examples. KW - Mathematics KW - Connection matrix KW - Index theorems LA - English ER -