Knot polynomials as new tool for turbulence research

In recent decades there has been overwhelming evidence that vorticity tends to get concentrated to form coherent structures, such as vortex filaments and tubes (the “sinews of turbulence”), in both classical and quantum fluids. In the case of vortex tangles, structural complexity methods have proven to be useful to investigate and establish new relations between energy, helicity and complexity [R1]. Indeed, this approach can be pursued further by introducing knot polynomials. By using a suitable transformation in terms of helicity, it has been recently shown [R2] that the standard Jones polynomial of knot theory can be interpreted as a new invariant of topological fluid mechanics. By briefly reviewing this work, we show how to compute this invariant for some simple, but non-trivial topologies. Then, by considering the standard decomposition of the helicity of a vortex filament in terms of writhe and twist, we focus on geometric aspects of vortex reconnection, a key feature in real vortex dynamics, and present a recent result [R1] on the conservation of writhe under reconnection. For dissipative systems, this means that any deviation from helicity conservation is entirely due to twist, inserted or deleted locally at the reconnection site. This result has important implications for helicity and energy considerations. We conclude the talk with some speculations on future work, by showing how a combination of these results may lead to the development of a novel approach to investigate fundamental aspects of turbulence research.
\begin{thebibliography}{1} \bibitem{R1} Ricca, R.L. (2009) Structural complexity and dynamical systems. In Lectures on Topological Fluid Mechanics (ed. R.L. Ricca), pp. 169-188. Springer-CIME Lecture Notes in Mathematics 1973. Springer-Verlag. \bibitem{R2} Liu, X., Ricca, R.L. (2012) The Jones polynomial for fluid knots from helicity. J. Phys. A: Math. and Theor., 45, 205501. \bibitem{R3} Laing, C.E., Ricca, R.L. and Sumners, DeW. L. (2013) Conservation of writhe and helicity under reconnection. PNAS, in preparation. \end{thebibliography}