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 $\mathrm{TP}(\gamma) = \frac1{2^{q}q}\iint\limits_{\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}} \frac{\mathrm d x\,\mathrm d y}{\mathbf{r}_{\gamma}(x,y)^q}, \qquad q>2,$ where $$\mathbf{r}(x,y)\in[0,\infty]$$ 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.

# VIDEO

## References

 [B] Sören Bartels. A simple scheme for the approximation of the elastic flow of inextensible curves. IMA J. Numer. Anal., 33:1115–1125, 2013. [ preprint | doi ] [BFRS] Sören Bartels, Philipp Falk, Philipp Reiter, Patrick Schoen. A tool for relaxing knots and embedded curves. In preparation. [BaR] Sören Bartels and Philipp Reiter. Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves. Submitted, 2018. [ preprint ] [BlR] Simon Blatt and Philipp Reiter. Regularity theory for tangent-point energies: The non-degenerate sub-critical case. Adv. Calc. Var., 8(2):93–116, 2015. [ arXiv | doi ] [BRR] Sören Bartels, Philipp Reiter, and Johannes Riege. A simple scheme for the approximation of self-avoiding inextensible curves. IMA Journal of Numerical Analysis, 2017. [ preprint | doi ] [GM] Oscar Gonzalez and John Maddocks. Global curvature, thickness, and the ideal shapes of knots. Proc. Natl. Acad. Sci. USA, 96(9):4769–4773 (electronic), 1999. [ open access ] [SvdM] Paweł Strzelecki and Heiko von der Mosel. Tangent-point self-avoidance energies for curves. J. Knot Theory Ramifications, 21(5):1250044, 2012. [ arXiv | doi ]