Henry Adams (8/30/21): Vietoris-Rips complexes of hypercube graphs

Questions about Vietoris-Rips complexes of hypercube graphs arise naturally from problems in genetic recombination, and also from Kunneth formulas for persistent homology with the sum metric. We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters. In more detail, let Q_n be the vertex set of the hypercube graph with 2^n vertices, equipped with the shortest path metric. Equivalently, Q_n is the set of all binary strings of length n, equipped with the Hamming distance. The Vietoris-Rips complex of Q_n at scale zero is 2^n points, and the Vietoris-Rips complex of Q_n at scale one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris-Rips complex of Q_n at scale two is homotopy equivalent to a wedge sum of 3-spheres, and furthermore we provide a formula for the number of 3-spheres as a function of n. Many questions about the Vietoris-Rips complexes of hypercube graphs at larger scale parameters remain open. We describe these questions, and pose some conjectures motivated by homology computations in Ripser++. Joint work with Michal Adamaszek.