Topological linking determines elasticity in limited valence networks.

Understanding the relationship between the microscopic structure and topology of a material and its macroscopic properties is a fundamental challenge across a wide range of systems. Here we investigate the viscoelasticity of DNA nanostar hydrogels-a model system for physical 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. Below the overlapping concentration, the fraction of branching points and the pore size determine the high-frequency elasticity of these physical gels. At higher concentrations, we discover that this elastic response is dictated by the abundance of topological links between looped motifs in the gel. Our findings highlight the emergence of 'topological elasticity' as a previously overlooked mechanism in generic network-forming liquids and gels and inform the design of topologically controllable material behaviours.