Abstract
A fourth-order topological invariant determined by the geometrical position of flux tubes linked with an even number of linkings is considered in ideal MHD. The topological invariant is determined by the properties of a surface bounded by flux tubes. For the Seifert surface it coincides with the Arf invariant. In particular, this topological invariant distinguishes “the Borromean rings” (three linked rings no two of which link each other) from three unlinked and unknotted rings.
Topological evolution of magnetic (or vortex) flux tubes in the limit of small diffusivity within a time such that helicity is conserved can be formalized as “a bordism”, an orientable surface without self-intersections in 4-dimensional space (3-space plus time) bounded by the set of flux tubes initially and at any subsequent time. The “births” and “deaths” of flux tubes correspond to the maxima and minima of the bordism, the reconnections are the saddle points of the bordism. Two sets of flux tubes belong to the same framed bordism if and only if they have the same helicities.
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© 1992 Springer Science+Business Media Dordrecht
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Akhmet’ev, P., Ruzmaikin, A. (1992). Borromeanism and Bordism. In: Moffatt, H.K., Zaslavsky, G.M., Comte, P., Tabor, M. (eds) Topological Aspects of the Dynamics of Fluids and Plasmas. NATO ASI Series, vol 218. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3550-6_13
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DOI: https://doi.org/10.1007/978-94-017-3550-6_13
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