  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.
