Institute of Geometry

The study of knot energies was initiated by the work of Moffatt and further developed by him following Arnold's work of 1974. The first discrete energy of knots were produced by W. Fukuhara in 1988.

A functional on the space of all knots is said to be an energy functional if it is bounded below, C¹-continuous, and tends to infinity while the knot tends to some ``knot'' with one point of a double transversal self-intersection. Gradient flows of such functionals bring arbitrary knots to some so-called perfect critical knots. Besides, for some energies it is conjectured, that the corresponding perfect critical knots are unique for any connected component of the space of all knots. So there is a hope that the set of perfect knots is a complete knot invariant. We found an integral equations on knots with critical functionals of energies, for a certain class of energies and introduced the Mm-energy of knots. This energy is good for numerical calculation of geometrical shapes of critical knots.

One of the most beautiful and significant energy functionals is Möbius energy. Möbius energy was discovered by J. O'Hara in 1991. Further investigations of Möbius energy properties were made by M. H. Freedman, Z.-H. He, and Z. Wang, they introduced variational principles, studied conformal properties, and found some upper estimates for minimal possible energy values for knots with a given crossing number. The notion of Möbius energy was generalized also to the case of the embedded graphs.

Jointly with A. B. Sossinski we studied a new type of energy functionals that are applied to planar knot diagrams. In the future we are aiming to study planar and space minimizers for knots and to find estimates on critical energies via some knot invariants.

Between 2008 and 2013, work on this project was carried out at TU Graz.

• S. Avvakumov, O. Karpenkov, and A. Sossinsky. Euler elasticae in the plane and the Whitney-Graustein theorem. Russ. J. Math. Phys. 20/3 (2013), 257-267. [MR], [doi].
• O. Karpenkov and A. Sossinsky. Energies of knot diagrams. Russian J. Math. Physics 18 (2011), 306-317. [doi].
• O. Karpenkov. The Möbius energy of graphs. Math. Notes 79/1 (2006), 134-138, Russian version: Mat. Zametki 79 (2007), 146-149. [Zbl], [MR], [doi], [arxiv:math-ph/0509055].
• O. Karpenkov. Energy of a knot: some new aspects. In Fundamental mathematics today, pages 214-223. Nezavis. Mosk. Univ., Moscow, 2003. [Zbl], [MR].
• O. Karpenkov. Energy of a knot: variational principles. Russ. J. Math. Phys. 9/3 (2002), 275-287. [Zbl], [MR].

• J. O'Hara. Energy of knots and conformal geometry, volume 33 of Series on Knots and Everything. World Scientific Publishing Co. Inc., River Edge, NJ, 2003. ISBN 981-238-316-6. [MR], [doi].
• J. M. Sullivan. Approximating ropelength by energy functions. In Physical knots: knotting, linking, and folding geometric objects in R³ (Las Vegas, NV, 2001), volume 304 of Contemp. Math., pages 181-186. Amer. Math. Soc., Providence, RI, 2002. [MR].
• M. H. Freedman, Z.-X. He, and Z. Wang. Möbius energy of knots and unknots. Ann. of Math. (2) 139/1 (1994), 1-50. [MR], [doi].
• J. O'Hara. Energy of a knot. Topology 30/2 (1991), 241-247. [MR], [doi].
• S. Fukuhara. Energy of a knot. In A fete of topology, pages 443-451. Academic Press, Boston, MA, 1988. [MR].
• V. I. Arnol'd. The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5/4 (1986), 327-345, Selected translations. [MR].
• H. K. Moffatt. Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology, Part 1. Fundamentals. Journal of Fluid Mechanics 159 (1985), 359-378. [doi].
• H. K. Moffatt. The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics 35 (1969), 117-129. [doi].