Entangled states are ubiquitous amongst fibrous materials, whether naturally occurring (keratin, collagen, DNA) or synthetic (nanotube assemblies, elastane). A key mechanical characteristic of these systems is their ability to reorganise in response to external stimuli, as implicated in e.g. hydration-induced swelling of keratin fibrils in human skin. During swelling, the curvature of individual fibres changes to give a cooperative and reversible structural reorganisation that opens up a pore network. The phenomenon is known to be highly dependent on topology, even if the nature of this dependence is not well understood: certain ordered entanglements (`weavings) can swell to many times their original volume while others are entirely incapable of swelling at all. Given this sensitivity to topology, it is puzzling how the disordered entanglements of many real materials manage to support cooperative dilation mechanisms. Here we use a combination of geometric and lattice-dynamical modelling to study the effect of disorder on swelling behaviour. The model system we devise spans a continuum of disordered topologies and is bounded by ordered states whose swelling behaviour is already known to be either vanishingly small or extreme. We find that while topological disorder often quenches swelling behaviour, certain disordered states possess a surprisingly large swelling capacity. Crucially, we show that the extreme swelling response previously observed only for certain specific weavings can be matched---and even superseded---by that of disordered entanglements. Our results establish a counterintuitive link between topological disorder and mechanical flexibility that has implications not only for polymer science but also for our broader understanding of collective phenomena in disordered systems.
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