Characterize the converse relation between elastic limits and ropelength minimizers

Establish whether every strong limit of a bending-minimizing sequence in a fixed tame knot class is obtained as a limit of global minimizers of the ropelength-regularized energies.

Background

The paper proves that every family of global minimizers of the ropelength-regularized functional has bending energy converging to the infimum and that every elastic-limit subsequence converges strongly in H² to a bending minimizer. It does not prove the converse selection statement: namely, whether every strong limit arising from arbitrary bending-minimizing sequences can be realized by ropelength minimizers as the regularization parameter tends to zero. Resolving this would clarify the relationship between the broader class of elastic limits and the particular limits selected by ropelength regularization.

References

The converse selection statement, that every such limit is obtained from ropelength minimizers, has not been proved.

— Elastic knots: compactness, contact, and variational structure  (2610.26823 - Reiter et al., 19 Oct 2026) in Remark following Proposition 5.2 in Section 4, “Strong limits and vanishing self-avoidance penalties”