An elastic knot of a tame knot class is a limiting closed curve obtained by minimizing bending energy at
fixed curve length while the imposed thickness tends
to zero. In the thickness-constrained formulation, the bending energy of unit-length,
arc length-parametrized embeddings 
in the Sobolev
space 
is minimized:
 |
where 
is the curvature vector. The thickness is the
infimum of the circumradii
of all triples of distinct points on the curve, with collinear
triples assigned infinite circumradius (Gonzalez
and Maddocks 1999). An elastic knot is a 
limit of global minimizers subject
to thickness at least 
along a sequence 
(Miura 2026).
The limiting curve may have self-intersections, even though every approximating curve is an embedding. Miura (2026) proved that every elastic knot of any twist knot class other than the unknot or trefoil knot is nonplanar. In particular, this refutes the proposed planar limiting configuration for the figure eight knot.
