Understanding how microscopic structure and topology shape macroscopic material properties is a central challenge across many fields [1–3]. In this talk, I will introduce a topological framework based on the linking of loops within limited valence networks, in which nodes are constrained to have a restricted number of connections. I will first discuss DNA nanostar hydrogels [4], where experiments and molecular simulations reveal highly interpenetrated, loop-rich structures [1]. At sufficiently high concentrations, topological linking becomes a dominant contribution to elasticity, establishing a direct connection between network topology and mechanical response. I will then show how similar ideas can be extended far beyond soft DNA networks. In amorphous ice, pressure drives a transformation between low- and high-density states accompanied by a dramatic reorganization of network topology and topological linking [5]. More recently, we have applied this framework to experimentally compatible atomistic models of silica, where linking distinguishes mechanically different amorphous and crystalline phases, while spectral analysis of the linking network reveals signatures of structural order [6]. Together, these results suggest that topological linking provides a common language for understanding structure–property relationships across network-forming materials, from programmable soft matter to materials under extreme conditions. References [1] G. Palombo et al, Nature Materials, 24, 454-461 (2025). [2] A. Neophytou et al, Nature Physics, 18, 1248-1253 (2022). [3] M. Bantawa et al, Nature Physics, (2023). [4] N. Conrad et al, PNAS, 116 (15), 7238-7243 (2019). [5] Fosado et al, Phys. Rev. Lett., 133, 266102 (2024). [6] Fosado et al, arXiv:2608.28365 (2026).