  Understanding the connection between a material's microscopic network
topology and its macroscopic properties is a fundamental open challenge across
a wide range of physical systems. Here, we investigate the elasticity of DNA
nanostar hydrogels - a model system for networks with limited valence - by
coupling rheology measurements, confocal imaging and molecular dynamics
simulations. We discover that these networks display a large degree of
interpenetration and that loops within the network are topologically linked,
forming a percolating network-within-network structure. Strikingly, we discover
that the onset of topological links between shortest loops fully determines the
elasticity of these physical gels. We thus argue that the elasticity of limited
valence gels is not dictated by their fractal microstructure, but by the
topology of looped motifs. Our findings highlight the emergence of "topological
elasticity" as a previously overlooked mechanism in generic network-forming
liquids and physical gels and will inspire the design of
topologically-controllable material behaviours.
