By making simple, heuristic assumptions, a new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial expressed in terms of kinetic helicity. Then, for the sake of illustration, explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knot and for the Whitehead link via the figure-of-eight knot are considered. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. While this heuristic approach extends the use of helicity in terms of linking numbers to the much richer context of knot polynomials, it gives also rise to new interesting problems in mathematical physics and it offers new tools to perform real-time numerical diagnostics of complex flows.

Liu, X., Ricca, R. (2013). Tackling fluid structures complexity by the Jones polynomial. In H.K. Moffatt, K. Bajer, Y. Kimura (a cura di), IUTAM Symposium on Topological Fluid Dynamics: Theory and Applications (pp. 175-182). Elsevier [10.1016/j.piutam.2013.03.021].

Tackling fluid structures complexity by the Jones polynomial

RICCA, RENZO
2013

Abstract

By making simple, heuristic assumptions, a new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial expressed in terms of kinetic helicity. Then, for the sake of illustration, explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knot and for the Whitehead link via the figure-of-eight knot are considered. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. While this heuristic approach extends the use of helicity in terms of linking numbers to the much richer context of knot polynomials, it gives also rise to new interesting problems in mathematical physics and it offers new tools to perform real-time numerical diagnostics of complex flows.
Capitolo o saggio
vortex knots and links; helicity; Jones polynomial; topological fluid dynamics; structural complexity
English
IUTAM Symposium on Topological Fluid Dynamics: Theory and Applications
Moffatt, HK; Bajer, K; Kimura, Y
2013
7
Elsevier
175
182
Liu, X., Ricca, R. (2013). Tackling fluid structures complexity by the Jones polynomial. In H.K. Moffatt, K. Bajer, Y. Kimura (a cura di), IUTAM Symposium on Topological Fluid Dynamics: Theory and Applications (pp. 175-182). Elsevier [10.1016/j.piutam.2013.03.021].
reserved
File in questo prodotto:
File Dimensione Formato  
IUTAMproc13_2.pdf

Solo gestori archivio

Dimensione 396.31 kB
Formato Adobe PDF
396.31 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/64472
Citazioni
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
Social impact