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Imagine a knotted wire with electric charge distributed uniformly along it. What do the equipotential surfaces around it look like? The animation below, by my student Max Lipton, shows one example. We’re also studying how many zeros exist in the electric field around the knot. https://twitter.com/maxematician/status/1263980938331127811
Max Lipton @Maxematician posted a preprint about charged knots last August. It includes a nifty lower bound on the number of equilibrium point around the knot in terms of a certain knot invariant. https://arxiv.org/abs/1908.01942
Just to be clear, our knots are rigid. We are interested in the zeros in the electric field around them. Other people have considered charged knots that change shape (but not their knot type) as they lower their energy by deforming under self-repulsion.
An equipotential surface for a charged trefoil knot, courtesy of my student Max Lipton @Maxematician: https://twitter.com/maxematician/status/1264365588333367303?s=21
Another equipotential surface for a charged knot, by @Maxematician. Click once on the gif to enlarge it, and another time to make it stop rotating so you can inspect it closely. https://twitter.com/maxematician/status/1264714152909684736?s=21
An equipotential surface around a “square knot” with electric charge distributed uniformly along its length, by @Maxematician. https://twitter.com/maxematician/status/1265055125414055936?s=21
And here is an equipotential surface around a granny knot; compare it to the square knot above in this thread. https://twitter.com/maxematician/status/1264974745155624961?s=21
