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On the derivation of the HOMFLYPT polynomial invariant for fluid knots

Published online by Cambridge University Press:  14 May 2015

Xin Liu  and
Renzo L. Ricca
Xin Liu
Affiliation:
Beijing–Dublin International College and Institute of Theoretical Physics, Beijing University of Technology, 100 Pingleyuan, Beijing 100124, PR China
Renzo L. Ricca*
Affiliation:
Department of Mathematics and Applications, University of Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy
*
†Email address for correspondence: renzo.ricca@unimib.it
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Abstract

By using and extending earlier results (Liu & Ricca, J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.

JFM classification

Mathematical Foundations: Mathematical Foundations Mathematical Foundations: Topological fluid dynamics

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 773 , 25 June 2015 , pp. 34 - 48
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© 2015 Cambridge University Press 

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