## Analysis of Quantised Vortex TangleIn this thesis, the author develops numerical techniques for tracking and characterising the convoluted nodal lines in three-dimensional space, analysing their geometry on the small scale, as well as their global fractality and topological complexity---including knotting---on the large scale. The work is highly visual, and illustrated with many beautiful diagrams revealing this unanticipated aspect of the physics of waves. Linear superpositions of waves create interference patterns, which means in some places they strengthen one another, while in others they completely cancel each other out. This latter phenomenon occurs on 'vortex lines' in three dimensions. In general wave superpositions modelling e.g. chaotic cavity modes, these vortex lines form dense tangles that have never been visualised on the large scale before, and cannot be analysed mathematically by any known techniques. |

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### Contents

1 | |

2 Numerical Methods | 45 |

3 Geometry and Scaling of Vortex Lines | 75 |

4 Topological Methods | 109 |

5 Knotting and Linking of Vortex Lines | 143 |

6 Conclusions | 189 |

Appendices | 193 |

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3-sphere 3-torus Alexander polynomial algorithm analysis analytic approximately arclength average behaviour calculation Cartesian closed loops coloured components correlation curvature and torsion degenerate eigenfunctions detected diagram discussed distribution eigenfunctions eigenfunctions of energy example exponential field fractal Gaussian random geometrical gradient grid cell hyperbolic volume hyperspherical infinite integral intersection invariants investigate Jones polynomial knot invariants knot type knots and links lattice lengthscale linking number minimal crossing number nodal lines non-trivial homology NTH lines octant parameterisation periodic boundaries persistence length phase singularities Phys physical plane waves plotted polygon polymer prime knots projection quantised quantities quantum harmonic oscillator radius random walk random wave model recovered region Reidemeister moves resampling resolution Sect shows side length simulations space curves statistics three dimensions three-dimensional topological torus trefoil knot two-dimensional unknot unknot probability vortex curves vortex lines vortex loops vortex segments vortex tangle vortices wave chaos wavelength wavevectors writhe