Vortex knots in tangled quantum eigenfunctions

Alexander Taylor, Mark Dennis

Research output: Contribution to journalArticle (Academic Journal)peer-review

30 Citations (Scopus)
429 Downloads (Pure)


Tangles of string typically become knotted, from macroscopic twine down to long-chain macromolecules such as DNA. Here, we demonstrate that knotting also occurs in quantum wavefunctions, where the tangled filaments are vortices (nodal lines/phase singularities). The probability that a vortex loop is knotted is found to increase with its length, and a wide gamut of knots from standard tabulations occur. The results follow from computer simulations of random superpositions of degenerate eigenstates of three simple quantum systems: a cube with periodic boundaries, the isotropic three-dimensional harmonic oscillator and the 3-sphere. In the latter two cases, vortex knots occur frequently, even in random eigenfunctions at relatively low energy, and are constrained by the spatial symmetries of the modes. The results suggest that knotted vortex structures are generic in complex three-dimensional wave systems, establishing a topological commonality between wave chaos, polymers and turbulent Bose–Einstein condensates.
Original languageEnglish
Article number12346
Number of pages6
JournalNature Communications
Publication statusPublished - 29 Jul 2016

Structured keywords



  • statistical physics
  • knot theory
  • quantum chaology
  • applied topology


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