# Cuts for 3-D magnetic scalar potentials: visualizing unintuitive surfaces arising from trivial knots

## TL;DR

This paper explores the visualization of complex surfaces arising from trivial knots in magnetic scalar potentials, introducing an energy minimization approach to compute cuts with topologically unintuitive features.

## Contribution

It presents a novel analysis of cuts for trivial loops, highlighting topologically complex surfaces and proposing an energy functional minimization algorithm for their computation.

## Key findings

- Unknotted curves can bound high genus surfaces in their convex hull.
- A conformally invariant functional can be minimized to compute cuts.
- The approach reveals unintuitive topological features of magnetic scalar potential cuts.

## Abstract

A wealth of literature exists on computing and visualizing cuts for the magnetic scalar potential of a current carrying conductor via Finite Element Methods (FEM) and harmonic maps to the circle. By a cut we refer to an orientable surface bounded by a given current carrying path (such that the flux through it may be computed) that restricts contour integrals on a curl-zero vector field to those that do not link the current-carrying path, analogous to branch cuts of complex analysis. This work is concerned with a study of a peculiar contour that illustrates topologically unintuitive aspects of cuts obtained from a trivial loop and raises questions about the notion of an optimal cut. Specifically, an unknotted curve that bounds only high genus surfaces in its convex hull is analyzed. The current work considers the geometric realization as a current-carrying wire in order to construct a magnetic scalar potential. Moreover, we consider the problem of choosing an energy functional on the space of maps, suggesting an algorithm for computing cuts via minimizing a conformally invariant functional utilizing Newton iteration.