The modern legacy of Tait’s knot theory

Following Lord Kelvin’s attempt to establish a vortex atom theory of matter (a proper topological field ‘theory of everything’ ante litteram), Peter Guthrie Tait undertook the immense task of studying and classifying knots and links, thus contributing to the origin of the mathematics of knot theory. His work remained little known until its re-discovery by J.H. Conway in the ‘70s and the subsequent work on the derivation of knot polynomial invariants culminated with the Fields Medal to V.F.R. Jones (1990). Meanwhile, a new form of theory of everything emerged with the work (crowned also by a Fields Medal in 1990) by E. Witten, encapsulating the Jones polynomials into the Chern-Simons theory of strings, that led to the modern topological quantum field theory. But it is only in the last decades that knot theory has permeated the real world. First, with the topological interpration of helicity (a conserved quantity of ideal fluid mechanics) by H.K. Moffatt (1969), where vortex dynamics and plasma physics become the playground of knotted fields, paving the way to the recent realisation and observation of optical knots, superfluid knots, Hopfions, Skyrmions and all sorts of knotted defects in liquid crystals and condensates (work that has ramified into DNA topology, polymer physics and quantum computation). Tait’s abstract world is every day more real and alive all around us, in our physical world.

[1] Ricca, R.L. & Liu, X. (Eds.) 2024 Knotted Fields. Lecture Notes in Mathematics 2344. Springer-Nature, Switzerland.
[2] Tubiana, L. et al. (2024) Topology in soft and biological matter. Physics Reports 1075, 1-137.
[3] Ricca, R.L 2025 Topological fluid dynamics and knotted fields. In Encyclopedia of Mathematical Physics (ed. R. Szabo & M. Bojowald) 4, pp. 245-255. Academic Press (2nd edition).

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