Determine whether the conditional obstruction occurs for higher bridge index

Determine whether there exists a tame knot class with bridge index a≥3 satisfying e_K=(2πa)^2 and braid(K)>a.

Background

Under the saturation condition e_K=(2πa)2 and the strict inequality braid(K)>a, the paper derives conditional obstructions: the full reaction of every normalized bending-minimizing closure curve or elastic limit cannot be a finite Radon measure, and such curves cannot belong to W{3,1}. The authors note that no knot class satisfying these hypotheses is exhibited. They explain that the hypotheses are impossible for a=2, while their existence for a≥3 remains unresolved.

References

For a≥3 these remain conditional obstructions; the existence of a class satisfying their hypotheses is not settled here.

— Elastic knots: compactness, contact, and variational structure  (2610.26823 - Reiter et al., 19 Oct 2026) in Final paragraph of Section 8, “Directional crookedness and a regularity obstruction”