### Abstract

The topology of divergence-free fields is important in many parts of physics, e. g. in magnetohydrodynamics, plasma physics, hydrodynamics, superfluids etc. With the focus on applications in magnetohydrodynamics, our principal aim is the characterisation of magnetic fields by the means of invariants. In this report an introduction to the problem of finding higher order invariants is given. Then a third-order link integral of three magnetic fields is presented, which can be shown to be a topological invariant and therefore an invariant in ideal magnetohydrodynamics. This integral generalises the known third-order link invariant derived from the Massey triple product, which could only be applied to isolated flux tubes. As an example three magnetic fields not confined to flux tubes are given that possess a third-order linkage.

Original language | English |
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Title of host publication | Tubes, sheets and singularities in fluid dynamics |

Subtitle of host publication | proceedings of the NATO ARW, 2-7 September 2001, Zakopane, Poland |

Editors | K. Bajer, H. K. Moffatt |

Publisher | Kluwer Academic Publishers |

Pages | 151-156 |

Number of pages | 6 |

ISBN (Print) | 9781402009808, 1402009801 |

DOIs | |

Publication status | Published - 2002 |

### Publication series

Name | Fluid mechanics and its applications |
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Number | 71(3) |

ISSN (Print) | 0926-5112 |

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## Cite this

Mayer, C., & Hornig, G. (2002). A third-order topological invariant for three magnetic fields. In K. Bajer, & H. K. Moffatt (Eds.),

*Tubes, sheets and singularities in fluid dynamics: proceedings of the NATO ARW, 2-7 September 2001, Zakopane, Poland*(pp. 151-156). (Fluid mechanics and its applications; No. 71(3)). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-48420-X_22