New method for computation of fluid helicity: Knot polynomial invariants

2010/05/22 by Xin Liu, Liu, Xin
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Geometric Topology (math.GT) #Plasma Physics (physics.plasm-ph) #math.GT #physics.flu-dyn #physics.plasm-ph

paper · pdf · doi:10.48550/arxiv.1005.4153

15 pages, 8 figures. This paper has been withdrawn and will be re-posted after improvement

arxiv created 2010/07/28 · arxiv updated 2010/07/29

Abstract

A new algebraic method for computing helicity is developed, by discovering a relationship between helicity of fluid mechanics and algebraic polynomial invariants of knot theory. We have constructed a topological invariant t^H(L) for a link L of knots, where H is the helicity of a given fluid and t a formal constant. For oriented knotted vortex lines, t^H satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, t^H satisfies the skein relations of the Kauffman bracket polynomial. Our new algebraic method is to use skein relations to compute the helicity of a link L by algebraic recursion.

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