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Jack Binysh

I am a third year PhD Student with the Mathsys CDT, supervised by Gareth Alexander. My main research interest is in the application of ideas from geometry and topology to condensed matter physics.


J.Binysh, C. Whitfield, G. P. Alexander, Stable and Unstable Vortex Knots in Excitable Media, Phys. Rev. E 99, 012211 (2019)

J.Binysh, G. P. Alexander, Maxwell's Theory of Solid Angle and the Construction of Knotted Fields, J. Phys. A, 51, 38 (2018)

Conferences/Summer Schools:


  • Hot Topics: Shape and Structure of Materials - MSRI, University of California, Berkeley, 1st-5th Oct
  • YRM 2018, University of Southampton, 23rd-26th July (Talk)
  • Soft matter and topology - KIAS, Seoul, 2nd-6th July (Poster)

  • IOP Theory of Condensed Matter annual meeting, 7th June (Poster)
  • "Geometric and Topological Methods in Liquid Crystals" BLCS one day event, 3rd April


  • Growth, form and self-organisation, Newton Institute, University of Cambridge, 22nd Aug - 20th Dec


  • Physics by the Lake, Cumberland Lodge, June (Poster)
  • BAMC 2016, University of Oxford, May
  • British Liquid Crystal Society Annual Meeting, University of Edinburgh

Some publication excerpts, figures etc.:

Stable and Unstable Vortex Knots in Excitable Media

Abstract: 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 62 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.


Untangling dynamics of the (a) 18 unknots, (b) 17 trefoils and (c) 9 figure eight knots formed by performing single
strand crossings on the higher crossing number knot geometries of §III. (a) All unknots simplify to a unique round geometry without reconnection events. Length decrease is monotonic, however there is some variation; the geometry of one particularly slow decay is shown in the inset, displayed at times indicated by the solid markers. (b) All trefoil geometries simplify to a unique stable state, however there is greater variation across decays than for the unknots, with periods where knot length actively increases (boxed inset, circled markers). (c) Of the 9 tangled figure eights simulated, 7 settle rapidly to a stable state. However, over T = 20000 one example fails to converge and another converges only after going through prolonged periods of length increase, contraction and irregular ‘tumbling’ dynamics.


Geometries, vortex framings and twist distributions of our stable knots. Vector fields along curves indicate vortex
framings, and are shown at four successive times across a (approximate) vortex rotation period. Curvatures and torsions shown correspond to the T = 0 panels (there is slight intra-period variation) with the zero of arclength fixed to maximal curvature values. With the exception of the 41 knot for which there is no distinction, all knots shown are the ‘right handed’ chiral variant — they rotate in a right handed sense about their direction of motion (down the page).

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

Abstract: We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in R3. 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.


Figure: The structure of the solid angle function around a knotted curve, a–c) show level sets of the solid angle of spacing π/2, each of which forms a Seifert Surface for the knot with opacities on the near sides of the images reduced to reveal the inner structure of the solid angle. a) A twisted unknot. b, c) The Whitehead link (components in blue, green) from two viewing directions. d) A slice through the Whitehead link from the same direction as c). The local structure of ω about the knot is especially clear in d) — ω winds by 4π, and as we move away from the knot, curvature induced corrections cause the level sets of ω to bunch along the curve normal.


Figure: The solid angle function can be used to initialise knotted vortex lines in a variety of systems. Here we show a knotted nematic texture for disclinations forming the Borromean rings, as generated using the solid angle function. The surface corresponds to the set of points where the director has no z component, dz = 0; it is coloured according the xy-components. In a) the texture is planar and in b) it is fully three-dimensional (the curve defining the xy-winding through a second solid angle function is also indicated).


My undergraduate degree was an MPhys in Physics at the University of Oxford 2009-2013. My final year project was on "Quantum information in accelerated reference frames" and was supervised by Prof. Vlatko Vedral. It was awarded the BP Prize for Theoretical Physics project.


An english translation of the russian langauge paper : A. M. Pertsov, E. A. Ermakova, Mechanism of the drift of a spiral wave in an inhomogeneous medium,. Biofizika 33, 338-342 (1988). by the amazing Iliana Peneva



Contact me:


Center for Complexity Science
Zeeman Building
University of Warwick