# Magnetic helicity in multiply connected domains

## TL;DR

This paper extends the concept of magnetic helicity to multiply connected domains, highlighting how domain topology influences helicity's topological and physical interpretation in magnetohydrodynamics.

## Contribution

It generalizes the standard helicity definition to complex topologies and discusses the role of the Biot-Savart operator in simplifying helicity expressions.

## Key findings

- Helicity depends on domain topology in multiply connected regions.
- Biot-Savart operator simplifies helicity calculations.
- Topological effects influence the physical interpretation of helicity.

## Abstract

Magnetic helicity is a fundamental quantity of magnetohydrodynamics that carries topological information about the magnetic field. By `topological information', we usually refer to the linkage of magnetic field lines. For domains that are not simply connected, however, helicity also depends on the topology of the domain. In this paper, we expand the standard definition of magnetic helicity in simply connected domains to multiply connected domains in $\mathbb{R}^3$ of arbitrary topology. We also discuss how using the classic Biot-Savart operator simplifies the expression for helicity and how domain topology affects the physical interpretation of helicity.