  In this Letter we study the phase transition between amorphous ices and the
nature of the hysteresis cycle separating them. We discover that a topological
transition takes place as the system transforms from low-density amorphous ice
(LDA) at low pressures, to high-density amorphous ice (HDA) at high pressures.
Specifically, we uncover that the hydrogen bond network (HBN) displays
qualitatively different topologies in the LDA and HDA phases: the former
characterised by disentangled loop motifs, while the latter displaying
topologically complex and metastable Hopf-linked and knotted configurations. At
the phase transition, the transient opening of the HBN topological motifs
yields mechanical fragility on the macroscale. Our results provide a detailed
microscopic description of the topological nature of the phase transition and
the hysteresis cycle between amorphous ices. We argue that the topological
transition discovered in this work may not only improve our understanding of
amorphous ices but represent a generic mechanism for the densification of
network-forming materials.
