Sossinski A. Energy functionals and the normal forms of knots and plane curves (abstract)

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Speaker: Alexei Sossinski

Title: Energy functionals and the normal forms of knots and plane curves

Abstract:
I will talk about knot theory, which is a topic that always interested Sasha Vinogradov: he translated the book by Ralph Fox on the subject and wrote an important supplement to the book.

This talk is a brief survey of ongoing research on the energy of knots, but will include a survey my own earlier work [1], [2], [5] and joint work [3], [4] with my former pupils O. Karpenkov and S. Avvakumov. The goal of these papers is to devise an algorithm (based on minimizing an energy functional) that will classify curves on the plane with a small number of double points and knots with a small number of crossings by bringing them to normal form.

The talk will begin with the demonstration of mechanical experiments with wire knots, showing how they automatically switch from any position to normal form and computer experiments with plane curves and knots, showing how our algorithm actually takes them to normal form. Then I will explain our algorithm: for polygonal knots, it minimizes a functional that consists of a summand that tries to straighten out the curve locally and a summand that forbids self-intersections (and the subsequent crossing of one part of the curve by another). In practice, the algorithm always terminates, which is actually amazing, because its termination is not a determinstic fact (i.e., not a theorem) – it is due to probabilistic laws of nature.

Another unexpected result of our study is that the normal form is not always unique – for the eight knot there are two: they depend on the mechanical characteristics of the wire, and in the case of implementation of our algorithm, on the initial shape of the knot.

Recently, I have constructed an algorithm minimizing a functional which consists of three summands, the two summands used before and a third summand based on the writhe of the knot. All the computer experiments with this algorithm have shown that the obtained normal form is unique. This is confirmed by physical experiments with new models of wire knots possessing a torque force due to twisting the wire before the ends of the wire are connected.

Time permitting, I will briefly explain our work on plane curves, a byproduct of which was our solution of the Euler elasticae problem, discuss some biological applications of our study, and formulate some conjectures about the further development of our approach.

References:
[1] Mechanical normal forms of knots and flat knots, Russian J. Math. Physics, Vol.18, no.2, 216-226 (2011).

[2] Normal forms of twisted wire knots, Russian J. Math. Physics, Vol.19, no.3, 394-400 (2012).

[3] [ Jointly with S.Avvakumov, O.Karpenkov ] Euler elasticae in the plane and the Whitney-Graustein theorem, Russian J. Math. Physics 20 (3), 257267 (2013).

[4] [ Jointly with S. Avvakumov ] bringing closed polygonal curves in the plane to normal form via local moves, Math. Notes, 103 (3) 466-473 (2018)

[5] Normal forms of unknotted ribbons and DNA, Russian J. Math. Physics, 25, No. 2, 241-247 (2018)

Event: Diffieties, Cohomological Physics, and Other Animals, 13-17 December 2021, Moscow.
Alexandre Vinogradov Memorial Conference.