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Knotty fields – quantum-topology-knot theory

[“Knot theory” series]

Twisted fields
Credit: Pixabay/CC0 Public Domain

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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