Abstract
Since the days of Maxwell and Tait, it has been known that the Gauss linking integral fails to detect entanglements with equal numbers of oppositely signed crossings, such as the Borromean rings. As the helicity or Gauss integral is quadratic in the fluxes or vortex strengths, it measures second order linking. It is the lowest member of a hierarchy of linking integrals. A third order linking integral (based on the Massey product) describes the subtle interlocking of the Borromean rings. An entanglement of four rings — any pair or triple of which is unlinked — is presented, together with a fourth order linking integral which distinguishes the delicate tangling from four unlinked rings. Links are close relatives of braids and the hierarchy of linking integrals now becomes a hierarchy of winding numbers. Detailed examples are given for n = 3 and 4 and the extension to n orders is sketched in the conclusions.
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© 1992 Springer Science+Business Media Dordrecht
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Evans, N.W., Berger, M.A. (1992). A Hierarchy of Linking Integrals. 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_12
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DOI: https://doi.org/10.1007/978-94-017-3550-6_12
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