  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.
