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Knotty fields – quantum-topology-knot theory

[“Knot theory” series]

torus-knot 3D
Apple’s Grapher

Regarding the topology of fundamental particles as spinors, like the electron, I prefer the term knot, rather than kink or twist or vortex [1]. Then I’m curious about ordering the complexity (à la generations) of these knots. And whether such knots are situated in additional dimensions.

Wiki: In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos.

So, these two recent articles caught my attention.

• Quanta Magazine > “How Complex Is a Knot? New Proof Reveals Ranking System That Works” by Leila Sloman, Writing Intern (May 18, 2022) – “Ribbon concordance” will let mathematicians compare knots by linking them across four-dimensional space.

Image caption: Two knots are concordant if they can be connected with a type of four-dimensional cylinder.

This winter, Ian Agol, a mathematician at the University of California, Berkeley, posted a six-page paper that proved Gordon‘s conjecture [re 4D ribbon concordance], giving mathematicians a new way to order knots by complexity.

Gordon’s conjecture is one of many in knot theory that attempt to organize the infinitely tangled universe of [closed loop] knots. At the heart of this project is the observation that you can drastically alter a knot’s appearance by twisting the strands or sliding them around. … Given drawings of knots, knot theorists try to figure out which ones are truly distinct, and which are different depictions of the same object.

In many cases, mathematicians resort to a less stringent concept called concordance. Concordance involves situating your three-dimensional knot in four-dimensional space. … [where] Two knots are concordant if they can be connected by a certain kind of imaginary [smooth] cylinder [which transforms one knot into the other].

Gordon thought that this property [of asymmetrical ribbon concordance] could be used to rank the knots against each other. If you need to add loops to the second knot in order to transform it into the first, it is effectively a less complicated knot. That creates an ordering of any set of knots that have ribbon concordances between them.

• > “Electrons in a crystal found to exhibit linked and knotted quantum twists” by Princeton University (May 20, 2022)

Image caption: Link diagram of the quantum electronic link in momentum (velocity) space observed in the topological Weyl magnet Co2MnGa, determined from advanced photoemission spectroscopy measurements. Credit: Ilya Belopolski and M . Zahid Hasan, Princeton Universit

… in an article published in Nature a Princeton-led team of physicists has discovered that electrons in quantum matter can link to one another in strange new ways. The work brings together ideas in three areas of science – condensed matter physics, topology, and knot theory

… as in the field of “quantum topology,” which seeks to explain an electron’s state as described by a property called its wave function.

“We’re studying properties related to the shape of the wave functions of electrons,” said Hasan [Eugene Higgins Professor of Physics, Princeton University].

The essential building block of this new frontier is a quantum mechanical structure known as a Weyl loop, which involves the winding of massless electron wave functions in a crystal.

… the case of Co2MnGa [crystals] … “Here instead we have linked loops—our newly discovered knotted topology is of a different nature and gives rise to different mathematical linking numbers,” said Tyler Cochran, a graduate student in Princeton’s Department of Physics and co-author of the new study.

This realization by Hasan’s team sparked fundamental questions about linked Weyl loops and brought together a team of experts from around the world in photoemission spectroscopy, mathematical topology, quantum material synthesis and first-principles quantum calculations to more deeply understand link topology and knotting in quantum matter.

Analysis of the experimental data revealed a counterintuitive object folded in on itself and wrapping across a higher-dimensional torus. “Understanding the object’s structure required a new bridge between quantum mechanics, mathematical topology and knot theory,” said Guoqing Chang, an author of the study who is now an assistant professor of physics at Nanyang Technological University in Singapore.


[1] In a sense that “particles” are “defects” where Grid energy flow is “frustrated” or knotted (tangled). Points where space-time and the quantum vacuum are not relaxed (à la vortices in a superfluid).


Winding (winding number)

Magnetic whirls

Topological magnets

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… it is now proving crucial to understanding the shapes of quantum waves formed by the electrons inside matter. These waves can form shapes such as vortices, knots and braids that give materials a variety of exotic properties. In 1983, Barry Simon was the first person to make the link between strange phenomena in materials and topology.

Symmetry → connection field → force of nature

… once you have a field, that field can have its own dynamics — it can bend and twist through space, typically in response to other fields that it interacts with. And when your gauge field starts twisting, you feel it as a force of nature.

2 thoughts on “Knotty fields – quantum-topology-knot theory

  1. Topological field sink

    Another test of the electron’s electric dipole moment measured no asymmetry at improved precision. A tabletop experiment using an innovative molecular trap. A key standard model property related to time symmetry.

    I still think the phrase “distribution of electric charge in the electron” might be misleading – rather than the “field symmetry around an electron,” the symmetry of field interaction with an electron. Particularly when this article mentions “poles” and particles that “flit in and out of the vacuum around it.”

    • > “Measurement of electron’s ‘shape’ dims hopes for discovery of new particles” by Adrian Cho (July 6, 2023) – Since an electron’s electric field is almost perfectly spherically symmetric, even a hint of a more complex dipole pattern would signal new physics.

    The result may also make it harder for theorists to explain how the infant universe generated more matter than antimatter, Doyle [physicist John Doyle, Harvard University, co-leader of a competing experiment that set the previous limit on a charge asymmetry][1] says.

    The experiment comes down to applying such [magnetic and electric] fields and looking for this tiny shift in the frequency of the electron’s twirling, Cornell [physicist Eric Cornell, National Institute of Standards and Technology (NIST) and the University of Colorado Boulder (CU Boulder), a co-author on the new study] says.

    Within the bond between the hafnium and fluorine atoms [inside an ionized molecule of hafnium fluoride], an electron experiences an electric field of 20 billion volts per centimeter. Crudely speaking, the physicists applied an external electric field to flip the molecular ion and its powerful internal field one way or the other relative to an applied magnetic field, while watching for any changes in the frequency at which the electron within twirled.


    [1] See this April 22, 2022 comment for “Feynman’s legacy — quantum originality.”

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  2. Spacetime fabric

    This video about Einstein’s general [vs. special] relativity equations is interesting. Not that I can grasp a higher-dimensional hypergraph, a geodesic ball of a Riemannian manifold, etc. [1] But that the derivation starts from some first principles and obtains Einstein’s formulation – an alternate approach..

    Also, this struck me for the non-vacuum derivation – “that particles can be treated as localized topological obstructions.” (Akin to my comments about so-called particles as “knots.”)

    • YouTube > The Last Theory > “How to derive general relativity from Wolfram Physics with Jonathan Gorard” (Sep 21, 2023) and hosted by Mark Jeffery – The structure of space-time in the presence of matter falls out of the hypergraph.


    One of the most compelling results to come out of the Wolfram Physics is Jonathan’s derivation of the Einstein equations from the hypergraph.

    Whenever I hear anyone criticize the Wolfram model for bearing no relation to reality, I tell them this: Jonathan Gorard has proved that general relativity can be derived from the hypergraph.

    In this excerpt from our conversation, Jonathan describes how making just three reasonable assumptions –

    • causal invariance
    • asymptotic dimension preservation
    • weak ergodicity

    – allowed him to derive the vacuum Einstein equations from the Wolfram model.

    In other words, the structure of space-time in the absence of matter more or less falls out of the hypergraph.

    And making one further assumption – that particles can be treated as localized topological obstructions – allowed Jonathan to derive the non-vacuum Einstein equations from the Wolfram model.

    In other words, the structure of space-time in the presence of matter, too, falls out of the hypergraph.

    It’s difficult to overstate the importance of this result.

    At the very least, we can say that the Wolfram model is consistent with general relativity.

    To state it more strongly: we no longer need to take general relativity as a given; instead, we can derive it from Wolfram Physics.


    [1] I have a general appreciation of the stress-energy tensor, but hardly these terms / concepts:

    • Hausdorff dimension
    • Geodesic balls, tubes & cones
    • Ricci scalar curvature
    • Ricci curvature tensor
    • Einstein equations
    • Einstein–Hilbert action
    • Relativistic Lagrangian density
    • Causal graph
    • Tensor rank
    • Trace

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