By using and extending earlier results (Liu &amp; 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.

Liu, X., Ricca, R. (2015). On the derivation of the HOMFLYPT polynomial invariant for fluid knots. JOURNAL OF FLUID MECHANICS, 773, 34-48 [10.1017/jfm.2015.231].

### On the derivation of the HOMFLYPT polynomial invariant for fluid knots

#### 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.
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Articolo in rivista - Review Essay
mathematical foundations; topological fluid dynamics; Mechanical Engineering; Mechanics of Materials; Condensed Matter Physics;
English
2015
773
34
48
reserved
Liu, X., Ricca, R. (2015). On the derivation of the HOMFLYPT polynomial invariant for fluid knots. JOURNAL OF FLUID MECHANICS, 773, 34-48 [10.1017/jfm.2015.231].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/90082`