Relaxing Embedded Curves

In general, the elastic flow of curves does not preserve the isotopy type of the initial configuration. Regularizing the bending energy by a small factor of a self-avoiding functional leads to situations close to self-contact. In order to avoid technical issues, we replace ropelength by a smooth functional.

Tangent-point potential

The tangent-point potential has been introduced by Gonzalez and Maddocks [GM] and investigated by Strzelecki and von der Mosel [SvdM]. It amounts to $$\mathrm{TP}(\gamma )=\frac{1}{2^{q}q}\underset{\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}}{\iint }\frac{\mathrm{d}x\mathrm{d}y}{{\mathbf{r}}_{\gamma }(x,y{)}^q},q>2,$$ where $\mathbf{r}(x,y)\in [0,\mathrm{\infty }]$ denotes the radius of the circle passing through $\gamma (x)$ and $\gamma (y)$ and being tangent to $\gamma$ at $\gamma (y)$. Numerically relevant is the fact that it involves only two integrations and that its derivative has an $L^1$ integrand [BlR].

Numerical simulation

The discretization of $\mathrm{TP}$ including error estimates has been derived in [BRR].

In [BaR] we consider a numerical scheme for the $H^2$ flow and derive a stability result. The latter is based on estimates for the second derivative of $\mathrm{TP}$ and a uniform bi-Lipschitz radius.

Experiments like the simulation shown on this page indicate a complex energy landscape.

Symmetry

Prescribing symmetry yields different evolutions and limit objects which are referred to as symmetric elastic knots, see [GiRvdM].

In the example shown on the left, we consider $D_3$ symmetry where $D_3$ denotes the dihedral symmetry group with six elements. The simulation has been produced using a modification of the scheme in [BaR].

The situation for more general knot classes and symmetries is wide open.

References