Institute of Geometry
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|Research project: Energies of knots and graphs|
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.