Topological constraints (TCs) between polymers determine the behaviour of complex fluids such as creams, oils and plastics. Most of the polymer solutions used every day life employ linear chains; their behaviour is accurately captured by the reptation and tube theories which connect microscopic TCs to macroscopic viscoelasticity. On the other hand, polymers with non-trivial topology, such as rings, hold great promise for new technology but pose a challenging problem as they do not obey standard theories; additionally, topological invariance -- i.e. the fact that rings must remain unknotted and unlinked if prepared so -- precludes any serious analytical treatment. Here we propose an unambiguous, parameter-free algorithm to characterise TCs in polymeric solutions and show its power in characterising TCs of entnagled rings. We analyse large-scale molecular dynamics (MD) simulations via persistent homology, a key mathematical tool to extract robust topological information from large datasets. This method allows us to identify ring-specific TCs which we call "homological threadings" (H-threadings) and to connect them to the polymers' behaviour. It also allows us to identify, in a physically appealing and unambiguous way, scale-dependent loops which have eluded precise quantification so far. We discover that while threaded neighbours slowly grow with the rings' length, the ensuing TCs are extensive also in the asymptotic limit. Our proposed method is not restricted to ring polymers and can find broader applications for the study of TCs in generic polymeric materials.

Accepted for publication in Physical Review Research. 10 pages, 4 figures in main text + 2 figures in appendix