Langer and Singer's question [LS] leads to a free obstacle problem that involves techniques at the interface of geometric analysis, low-dimensional topology, modeling, numerical analysis, and nonlinear optimization.

Elastic energy

Neglecting all effects of twist and shear, we consider a simplistic model by assuming that the shape of the wire is only influenced by the bending energy of its centerline $\gamma :\mathbb{R}/\mathbb{Z}\to {\mathbb{R}}^3$. The latter amounts to $$E_{\mathrm{bend}}(\gamma )=\underset{\mathbb{R}/\mathbb{Z}}{\int }{\kappa }_{\gamma }(s{)}^2\mathrm{d}s$$ where ${\kappa }_{\gamma }(s)$ denotes the scalar curvature at $\gamma (s)$ and $s$ is an arc-length parameter.

The model

In order to model the impermeability of the curve, we regularize the bending energy by a functional $\mathcal{R}$ that models self-avoidance. This means that its values blow up on sequences of embedded curves converging to a curve with a self-intersection. There are several candidates for this purpose.

As proposed in [vdM], we study the problem of minimizing the total energy $$E_{\vartheta }=E_{\mathrm{bend}}+\vartheta \mathcal{R}$$ within a given knot class. We expect that the limit of $E_{\vartheta }$ minimizers as $\vartheta \searrow 0$ does not depend on the choice of $\mathcal{R}$.

Employing the ropelength functional, i.e., the quotient of length over thickness, reflects the idea of a tube with a uniform radius. Thickness of a curve can be defined as the infimum over the radii of all circles passing through three distinct points of the curve [GM].

Two-bridge torus knots

The proof relies on a generalization of the Fáry–Milnor theorem [M] to the $C^1$ closure of a knot class which in turn bases on the existence of alternating quadrisecants for knotted curves due to Denne [D].

References