We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence.

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function on the 2-sphere and simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson’s correspondence.

We associate to any simplicial complex a filtration, starting from the discrete Morse complex and finishing at the matching complex. This leads to the definition of several homology theories, which we compute in a number of examples. We completely determine the graded object associated to this filtration in terms of the homology of simpler complexes.

We analyse proteins forming open-ended trefoil knots by introducing a topologically inspired statistical metric that measures their entanglement. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for such proteins.

It is often computationally expensive to determine topological features of the directed flag complex associated to a digraph. We introduce methods to measure how well graph pseudometrics capture the topology of a digraph, and apply them to various random digraph models.