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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: [email protected]
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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
Type
Papers
Information
Journal of Fluid Mechanics , Volume 773 , 25 June 2015 , pp. 34 - 48
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Copyright
© 2015 Cambridge University Press 

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References

Adams, C. 1999 The Knot Book. Freeman.Google Scholar
Baggaley, A. W., Barenghi, C. F., Shukurov, A. & Sergeev, Y. A. 2012 Coherent vortex structures in quantum turbulence. Europhys. Lett. 98, 26002.CrossRefGoogle Scholar
Barenghi, C. F., Ricca, R. L. & Samuels, D. C. 2001 How tangled is a tangle? Physica D 157, 197–206.CrossRefGoogle Scholar
Berger, M. A. & Field, G. B. 1984 The topological properties of magnetic helicity. J. Fluid Mech. 147, 133–148.CrossRefGoogle Scholar
Buck, S. & Simon, J. 2012 The spectrum of filament entanglement complexity and an entanglement phase transition. Proc. R. Soc. Lond. A 468, 4024–4040.Google Scholar
Călugăreanu, G. 1961 Sur les classes d’isotopie des nœuds tridimensionnels et leurs invariants. Czechoslovak Math. J. 11, 588–625.CrossRefGoogle Scholar
Cirtain, J. W., Golub, L., Winebarger, A. R., De Pontieu, B., Kobayashi, K., Moore, R. L., Walsh, R. W., Korreck, K. E., Weber, M., McCauley, P., Title, A., Kuzin, S. & DeForest, C. E. 2013 Energy release in the solar corona from spatially resolved magnetic braids. Nature 493, 501–503.CrossRefGoogle ScholarPubMed
Douady, S., Couder, Y. & Brachet, M. E. 1991 Direct observation of the intermittency of intense vorticity filaments in turbulence. Phys. Rev. Lett. 67, 983–986.CrossRefGoogle ScholarPubMed
Enciso, A. & Peralta-Salas, D. 2012 Knots and links in steady solutions of the Euler equation. Ann. Maths 175, 345–367.CrossRefGoogle Scholar
Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K. & Ocneanu, A. 1985 A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239–246.CrossRefGoogle Scholar
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2007 Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics. J. Fluid Mech. 592, 335–366.CrossRefGoogle Scholar
Kauffman, L. H. 1987 On Knots. Princeton University Press.Google Scholar
Kauffman, L. H. 2001 Knots and Physics. World Scientific.CrossRefGoogle Scholar
Kedia, H., Bialynicki-Birula, I., Peralta-Salas, D. & Irvine, W. T. M. 2013 Tying knots in light fields. Phys. Rev. Lett. 111, 150404.CrossRefGoogle ScholarPubMed
Kleckner, D. & Irvine, W. T. M. 2013 Creation and dynamics of knotted vortices. Nat. Phys. 9, 253–258.CrossRefGoogle Scholar
KnotAtlas: http://katlas.math.toronto.edu/wiki/Main\_Page.Google Scholar
KnotPlot: http://www.knotplot.com.Google Scholar
Kondaurova, L., L’vov, V., Pomyalov, A. & Procaccia, I. 2014 Structure of quantum vortex tangle in $^{4}$ He counterflow turbulence. Phys. Rev. B 89, 014502.CrossRefGoogle Scholar
Leadbeater, M., Samuels, D. C., Barenghi, C. F. & Adams, C. S. 2003 Decay of superfluid turbulence via Kelvin-wave radiation. Phys. Rev. A 67, 015601.CrossRefGoogle Scholar
Lickorish, W. B. R. & Millett, K. C. 1988 The new polynomial invariants of knots and links. Maths Mag. 61, 3–23.CrossRefGoogle Scholar
Liu, X. & Ricca, R. L. 2012 The Jones polynomial for fluid knots from helicity. J. Phys. A 45, 205501.CrossRefGoogle Scholar
Liu, X. & Ricca, R. L. 2013 Tackling fluid structures complexity by the Jones polynomial. In Topological Fluid Dynamics: Theory and Applications (ed. Moffatt, H. K., Bajer, K. & Kimura, Y.), Procedia IUTAM, vol. 7, pp. 175–182. Elsevier.Google Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.CrossRefGoogle Scholar
Moffatt, H. K. 1990 The energy spectrum of knots and links. Nature 347, 367–369.CrossRefGoogle Scholar
Moffatt, H. K. & Ricca, R. L. 1992 Helicity and the Călugăreanu invariant. Proc. R. Soc. Lond. A 439, 411–429.Google Scholar
Moisy, F. & Jiménez, J. 2004 Geometry and clustering of intense structures in isotropic turbulence. J. Fluid Mech. 513, 111–133.CrossRefGoogle Scholar
Mouri, H., Hori, A. & Kawashimab, Y. 2007 Laboratory experiments for intense vortical structures in turbulence velocity fields. Phys. Fluids 19, 055101.CrossRefGoogle Scholar
Przytycki, J. H. & Traczyk, P. 1987 Conway algebras and skein equivalence of links. Proc. Am. Math. Soc. 100, 744–748.CrossRefGoogle Scholar
Ricca, R. L. 2013 New energy and helicity lower bounds for knotted and braided magnetic fields. Geophys. Astrophys. Fluid Dyn. 107, 385–402.CrossRefGoogle Scholar
Ricca, R. L. 2014 Structural complexity of vortex flows by diagram analysis and knot polynomials. In How Nature Works (ed. Zelinka, I., Sanayei, A., Zenil, H. & Rössler, O. E.), Emergence, Complexity and Computation, vol. 5, pp. 81–100. Springer.CrossRefGoogle Scholar
Ricca, R. L. & Liu, X. 2014 The Jones polynomial as a new invariant of topological fluid dynamics. Fluid Dyn. Res. 46, 061412.CrossRefGoogle Scholar
Ricca, R. L. & Moffatt, H. K. 1992 The helicity of a knotted vortex filament. In Topological Aspects of the Dynamics of Fluids and Plasmas (ed. Moffatt, H. K., Zaslavsky, G. M., Comte, P. & Tabor, M.), pp. 225–236. Kluwer Academic.CrossRefGoogle Scholar
Shimokawa, K., Ishihara, K., Grainge, I., Sherratt, D. J. & Vazquez, M. 2013 FtsK-dependent XerCD-dif recombination unlinks replication catenanes in a stepwise manner. Proc. Natl Acad. Sci. USA 110, 20906–20911.CrossRefGoogle Scholar
Tsubota, M., Araki, T. & Nemirovskii, S. K. 2000 Dynamics of vortex tangle without mutual friction in superfluid $\text{}^{4}\text{He}$ . Phys. Rev. B 62, 11751–11762.CrossRefGoogle Scholar