Gareth P. Alexander

Three-Dimensional Active Defect Loops

We describe the flows and morphological dynamics of topological defect lines and loops in three-dimensional active nematics and show, using theory and numerical modeling, that they are governed by the local profile of the orientational order surrounding the defects. Analyzing a continuous span of defect loop profiles, ranging from radial and tangential twist to wedge \textpm 1/2 profiles, we show that the distinct geometries can drive material flow perpendicular or along the local defect loop segment, whose variation around a closed loop can lead to net loop motion, elongation, or compression of shape, or buckling of the loops. We demonstrate a correlation between local curvature and the local orientational profile of the defect loop, indicating dynamic coupling between geometry and topology. To address the general formation of defect loops in three dimensions, we show their creation via bend instability from different initial elastic distortions.

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Jack Binysh
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Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics

We describe the geometry of bend distortions in liquid crystals and their fundamental degeneracies, which we call {$\beta$} lines; these represent a new class of linelike topological defect in twist-bend nematics. We present constructions for smecticlike textures containing screw and edge dislocations and also for vortexlike structures of double twist and Skyrmions. We analyze their local geometry and global structure, showing that their intersection with any surface is twice the Skyrmion number. Finally, we demonstrate how arbitrary knots and links can be created and describe them in terms of merons, giving a geometric perspective on the fractionalization of Skyrmions.

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Jack Binysh
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Stable and Unstable Vortex Knots in Excitable Media

We study the dynamics of knotted vortices in a bulk excitable medium using the FitzHugh-Nagumo model. From a systematic survey of all knots of at most eight crossings we establish that the generic behaviour is of unsteady, irregular dynamics, with prolonged periods of expansion of parts of the vortex. The mechanism for the length expansion is a long-range `wave slapping’ interaction, analogous to that responsible for the annihilation of small vortex rings by larger ones. We also show that there are stable vortex geometries for certain knots; in addition to the unknot, trefoil and figure eight knots reported previously, we have found stable examples of the Whitehead link and $6_2$ knot. We give a thorough characterisation of their geometry and steady state motion. For the unknot, trefoil and figure eight knots we greatly expand previous evidence that FitzHugh-Nagumo dynamics untangles initially complex geometries while preserving topology.

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Jack Binysh
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Maxwell's Theory of Solid Angle and the Construction of Knotted Fields

We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in $\textbackslash mathbb\textbraceleft R\textbraceright\textasciicircum 3$. This is a purely geometric construction in which all of the properties of the entire knotted field derive from the geometry of the curve, and from projective and spherical geometry. We emphasise a fundamental homotopy formula as unifying different formulae for computing the solid angle. The solid angle induces a natural framing of the curve, which we show is related to its writhe and use to characterise the local structure in a neighborhood of the knot. Finally, we discuss computational implementation of the formulae derived, with C code provided, and give illustrations for how the solid angle may be used to give explicit constructions of knotted scroll waves in excitable media and knotted director fields around disclination lines in nematic liquid crystals.

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Jack Binysh
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