Relaxing Embedded Curves

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

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.

Symmetry

An elastic \((3,4)\) torus knot is (expected to be) a three-fold circle. Prescribing dihedral symmetry, we arrive at a union of three circles meeting tangentially with 120° angles.

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

[BFS]	Sören Bartels, Philipp Falk, Pascal Weyer. KNOTevolve – a tool for relaxing knots and inextensible curves. Web application, 2020. [ link ]
[BaR]	Sören Bartels and Philipp Reiter. Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves. Accepted for publication by Mathematics of Computation, 2021. [ arXiv \| doi ]
[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 ]
[GiRvdM]	Alexandra Gilsbach, Philipp Reiter, Heiko von der Mosel. Symmetric elastic knots. Preprint 2021. [ arXiv ]
[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 ]