The topology of Helmholtz domains

TL;DR

This paper classifies Helmholtz and weakly-Helmholtz domains in 3D topology, revealing their elementary structure and connections to knot theory, with implications for electromagnetism, fluid dynamics, and elasticity.

Contribution

It provides a comprehensive topological classification of Helmholtz domains and relates weakly-Helmholtz domains to homology boundary links in knot theory.

Findings

Helmholtz domains are essentially knotted handlebodies or complements of trivial links.

Weakly-Helmholtz domains include homology boundary links.

Topology of Helmholtz domains is quite elementary.

Abstract

The goal of this paper is to describe and clarify as much as possible the 3-dimensional topology underlying the Helmholtz cuts method, which occurs in a wide theoretic and applied literature about Electromagnetism, Fluid dynamics and Elasticity on domains of the ordinary space. We consider two classes of bounded domains that satisfy mild boundary conditions and that become "simple" after a finite number of disjoint cuts along properly embedded surfaces. For the first class (Helmholtz), "simple" means that every curl-free smooth vector field admits a potential. For the second (weakly-Helmholtz), we only require that a potential exists for the restriction of every curl-free smooth vector field defined on the whole initial domain. By means of classical and rather elementary facts of 3-dimensional geometric and algebraic topology, we give an exhaustive description of Helmholtz domains, realizing that their topology is forced to be quite elementary (in particular, Helmholtz domains with connected boundary are just possibly knotted handlebodies, and the complement of any non-trivial link is not Helmholtz). The discussion about weakly-Helmholtz domains is a bit more advanced, and their classification appears to be a quite difficult issue. Nevertheless, we provide several interesting characterizations of them and, in particular, we point out that the class of links with weakly-Helmholtz complements eventually coincides with the class of the so-called homology boundary links, that have been widely studied in Knot Theory.