Applications of knot theory in fluid mechanics

摘要: In this paper we present an overview of some recent results on applications knot theory in fluid mechanics, as part a new discipline called ‘topological mechanics’ (TFM). The choice the topics covered here is deliberately restricted to those areas that involve mainly combination ideal mechanics techniques and concepts, complemented with brief description other concepts have important systems. We begin concept topological equivalence flow maps, giving definition knotted linked flux-tubes. context Reidemeister’s moves are interpreted terms local actions flows performed structures. An old theorem Lichtenstein (1925) concerning isotopic evolution vortex structures Euler equations re-proposed discussed TFM for first time. Then, review relationship between helicity linking numbers magnetic relaxation linked, braided magnetohydrodynamics. (and under certain approximations given by so-called ‘localized induction’ structures) briefly examine interesting relationships integrability existence stability filaments shape torus knots. conclude electrically charged knots embedded viscous fluid, elastic strings braids energy levels information. Some simple bounds global geometric quantities presented 1. Modern developments mechanics. Knotted ubiquitous nature particular. Their scale lengths range from 10÷10m, tiny macromolecules, polymers, defect lines superfluid vortices, 10 ÷ 10m, eddies, filaments, tornadoes, up 10m gigantic flux-tubes, plasma loops arches 1991 Mathematics Subject Classification: Primary 76-02; Secondary 54M99. Financial support Leverhulme Trust kindly acknowledged. final form no version it will be published elsewhere.