Fig. 9: Several of the NSi cycles found by persistent homology for the compressed glasses.

(Not to scale). The birth scale b_i of a cycle corresponds to the radius at which the spheres, centered on each of the Si atom sites, first overlap to enclose a hole. The Si-Si distance between sites in a cycle \(\le 2\sqrt {b_i}\), so each cycle is represented by Si atoms (solid circles) joined by bars of length \(\le 2\sqrt {b_i}\). The death scale d_j of a cycle corresponds to the radius at which the spheres cover the enclosed hole such that the hole is eliminated. The coordinates (b_i, \(\ell _{ij}\)) give the birth and lifetime values of a cycle (in units of Ų) where \(\ell _{ij} = d_j - b_i\). The cycles are optimal, i.e., each cycle corresponds to a minimal path length around a hole³⁶.