The evolution of a \(7_7\) knot [BFRS]

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.

References

[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 ]