numerical-knot-untangling

Read the paper here.

Overview

This is a research project for untangling knots numerically, consisting of an implementation for the Möbius knot energy minimization.

Mobius energy

The Möbius energy is defined as such:

$$ \mathcal{E}(\gamma)=\iint _{M^{2}} \frac{1}{|\gamma(u)-\gamma(v) |^{2}}-\frac{1}{d(u,v)^{2}} , dv , du $$

where:

• $\gamma(u):M\to\mathbb{R}^{3}$ is a parameterized curve representing a knot.
• $u$ is the parameter and denotes a length along $\gamma$
• $d(u,v)$ denotes the shortest distance between $u$ and $v$ along the curve

Energy computation on 2D hourglass curve

Roughly, since the curve $\gamma$ is completely described by a list of vertices, we use the following system

$$ \gamma_t = -\nabla \mathcal{E}_\gamma $$

Since we are iterating over a discrete set $M$ of $s$ values representing the indices of our vertices, the discretized energy $\hat{\mathcal{E}}$ is expressed by approximate the double integral by a double summation.

The gradient of this summation is computed and implemented in mobius.py as the function Mobius_gradient.

Gradient flow approximation

Implements a simple forward Euler method with respect to the above system. The scheme includes an dynamic step size which bounds the spatial step size and helps with stability.

The main file is mobius.py which the Mobius energy gradient computation.
The file config.py contains the configuration parameters for the Mobius energy gradient computation such as .obj file and visualization parameters.
The file curve.py contains the geometry manipulation function utility functions such as reading obj files.

Results

Torus knot $T_{4,1}$ (untangles to the unknot)

Overhand (untangles to the trefoil)

Trefoil for reference:

Closed reef knot (untangles to two unknot components.)