A new method based on the use of the Jones polynomial, a well-known topological invariant of knot theory, is introduced to tackle and quantify topological aspects of structural complexity of vortex tangles in ideal fluids. By re-writing the Jones polynomial in terms of helicity, the resulting polynomial becomes then function of knot topology and vortex circulation, providing thus a new invariant of topological fluid dynamics. Explicit computations of the Jones polynomial for some standard configurations, including the Whitehead link and the Borromean rings (whose linking numbers are zero), are presented for illustration. In the case of a homogeneous, isotropic tangle of vortex filaments with same circulation, the new Jones polynomial reduces to some simple algebraic expression, that can be easily computed by numerical methods. This shows that this technique may offer a new setting and a powerful tool to detect and compute topological complexity and to investigate relations with energy, by tackling fundamental aspects of turbulence research.

Ricca, R., Liu, X. (2014). The Jones polynomial as a new invariant of topological fluid dynamics. FLUID DYNAMICS RESEARCH, 46, 061412 [10.1088/0169-5983/46/6/061412].

The Jones polynomial as a new invariant of topological fluid dynamics

RICCA, RENZO;
2014

Abstract

A new method based on the use of the Jones polynomial, a well-known topological invariant of knot theory, is introduced to tackle and quantify topological aspects of structural complexity of vortex tangles in ideal fluids. By re-writing the Jones polynomial in terms of helicity, the resulting polynomial becomes then function of knot topology and vortex circulation, providing thus a new invariant of topological fluid dynamics. Explicit computations of the Jones polynomial for some standard configurations, including the Whitehead link and the Borromean rings (whose linking numbers are zero), are presented for illustration. In the case of a homogeneous, isotropic tangle of vortex filaments with same circulation, the new Jones polynomial reduces to some simple algebraic expression, that can be easily computed by numerical methods. This shows that this technique may offer a new setting and a powerful tool to detect and compute topological complexity and to investigate relations with energy, by tackling fundamental aspects of turbulence research.
Articolo in rivista - Articolo scientifico
Fluid dynamics; Numerical methods; Topology; Vortex flow Algebraic expression; Jones polynomial; Linking numbers; Structural complexity; Topological complexity; Topological invariants; Turbulence research; Vortex circulation
English
2014
46
061412
reserved
Ricca, R., Liu, X. (2014). The Jones polynomial as a new invariant of topological fluid dynamics. FLUID DYNAMICS RESEARCH, 46, 061412 [10.1088/0169-5983/46/6/061412].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/64566
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