Hopf fibration and knots

Does anybody here know some good resources or explanations of Hopf fibration and knots with GA?

You could start with my recent paper on Villarceau Rotors, which contains a full description of the Hopf fibration, and links to an earlier paper on knots.

Leo Dorst


Definitely read @LeoDorst papers and related links (pun intended), especially his "3D conformal motions " and “Square Roots” papers.

Here also is a video of some knots wound around transforming hopf links I made with CGA
https://vimeo.com/62755547. A potentially useful description of how this works can be found on page 131 of my thesis (http://versor.mat.ucsb.edu/ArticulatingSpace.pdf), where I discuss a way to make a Hopf Fibration by finding the transformation that takes a Circle to its own Axis. Interestingly, polar Hopf links are orthogonal and commuting and so the (dual of) two hopf linked circles can be considered well-chosen (orthogonal and commuting) planes (e.g. commuting bivectors) around which one can weave a knot à la Dorst.

Also, I always found this article on Lorenz and Modular flows fascinating but beyond my understanding: http://www.josleys.com/articles/ams_article/Lorenz3.htm If I understand what you are up to @chakravala, you would probably get this modular flow stuff better than I do. Would love to see a GA explanation of this.


That’s correct @wolftype the modular flows are a central aspect of my own research directions. Thanks for sharing the links, I am working towards making some future contributions in this area.