# Analysis of Quantised Vortex Tangle by Alexander John Taylor

By Alexander John Taylor

In this thesis, the writer develops numerical thoughts for monitoring and characterising the convoluted nodal strains in three-d area, analysing their geometry at the small scale, in addition to their international fractality and topological complexity---including knotting---on the massive scale. The paintings is very visible, and illustrated with many attractive diagrams revealing this unanticipated point of the physics of waves. Linear superpositions of waves create interference styles, this means that in a few areas they improve each other, whereas in others they thoroughly cancel one another out. This latter phenomenon happens on 'vortex traces' in 3 dimensions. generally wave superpositions modelling e.g. chaotic hollow space modes, those vortex strains shape dense tangles that experience by no means been visualised at the huge scale sooner than, and can't be analysed mathematically via any identified concepts.

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A is the so-called Hopf link, and b the Whitehead link 24 1 Introduction Whitehead link. As with the example knots, neither representation may be contorted to the other, so they are topologically distinct. Links may be different by having individual components wind more about each other, or by adding more loops that are also topologically entangled with one or more of the others, or by changing the knot type of one or more individual loops. We have already seen the linking number as an example of a quantity that can distinguishes different links, but it is far from a perfect tool for this; it is easy to construct topologically distinct links that have the same linking number, and in fact the Whitehead link has linking number 0 just as would be the case for two unlinked curves.

In the final Chap. 5, we discuss the results of applying topological measures to our curves. As with their geometry this includes comparison with analytic results and other systems where possible, though there are far fewer known results on vortex tangle topology. We take particular interest in the differences between our systems of wave chaos, in which knotting and linking may be expected to be strongly affected by their different constraints, but whose differences are not directly predictable.

We have already hinted in Fig. 9 that this does not apparently pose a problem, as even relatively small numbers of wavevectors in superposition closely approximate the truly isotropic limit. It may even be the case that the zeros of the field are a better statistical approximation of the isotropic limit than other features such as the maxima of intensity, since the latter may depend more subtly on the normalisation of the power spectrum. 6). Under such a model it is possible to simulate some volume of a chaotic wavefield, to track the vortex lines within it (we discuss this in detail in Chap.