Evolution of multiple closed knotted curves in space

TL;DR

This work studies the evolution of multiple closed space curves in $\mathbb{R}^3$ under curvature- and torsion-driven flow with mutual nonlocal interactions, including a Biot-Savart-type forcing term. It combines a direct Lagrangian formulation with the theory of analytic semi-flows to prove local existence, uniqueness, and continuation of Hölder-smooth solutions, and introduces a flowing finite-volume discretization (along with a method-of-lines approach) to numerically solve the resulting nonlinear parabolic system. The authors provide rigorous well-posedness results and a robust numerical scheme, complemented by computational experiments using Biot-Savart-type nonlocal forces that generate knotted and linked curve dynamics in 3D. Overall, the paper delivers both a rigorous mathematical framework and practical algorithms for simulating complex 3D knot evolution, with implications for vortex dynamics and dislocation-loop modeling in applied settings.

Abstract

We investigate a system of geometric evolution equations describing a curvature and torsion driven motion of a family of 3D curves in the normal and binormal directions. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semi-flows, we are able to prove local existence, uniqueness, and continuation of classical Hölder smooth solutions to the governing system of non-linear parabolic equations modelling $n$ evolving curves with mutual nonlocal interactions. We present several computational studies of the flow that combine the normal or binormal velocity and considering nonlocal interaction.

Figures (7)

• Figure 3.1: Discretization of a segment of a 3D curve by the method of flowing finite volumes.
• Figure 4.1: Two linked circles a) and the Biot-Savart force vector field induced by $\Gamma^2$ acting on points of $\Gamma^1$, b).
• Figure 4.2: The Listing's 8-knot curve linked with a circle a) and an ellipse b)
• Figure 4.3: Evolution of the initial linked circles parameterized by (\ref{['eq-crvs1-explicit']}).
• Figure 4.4: Evolution of the initial linked circles parameterized by (\ref{['eq-crvs2-explicit']}).