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

Alex Stockrahm*, Valtteri Lahtinen, Jari J.J. Kangas, P. Robert Kotiuga

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


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.

Original languageEnglish
Pages (from-to)3200-3210
Number of pages11
JournalComputers and Mathematics with Applications
Issue number9
Publication statusPublished - 1 Nov 2019
MoE publication typeA1 Journal article-refereed


  • Homology
  • Magnetic fields
  • Visualization


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