Knot energy
In knot theory, a knot energy is a rule that assigns a number to each knot shape. A knot is a loop (a circle) embedded in three-dimensional space. We usually work with a space of nicely behaved curves and only consider those that don’t intersect themselves as valid knots.
A knot energy is a function that takes a knot and returns a real number (or infinity). It is called self-repulsive if, whenever you deform a knot to try to create a self-crossing, the energy goes to infinity. It also needs to be bounded below. With these properties, when you try to minimize the energy (using gradient descent), the knot’s topological type (its knot class) does not change.
The idea often comes from imagining the knot as a string with electrical charges along it: like charges repel, so the knot tends to spread out to lower its energy. However, the naive integral for such a model diverges, so a regularization is needed to make the energy finite.
Historically, energies for polygonal knots were studied by Fukuhara in 1987 and soon after by Sakuma. In 1988 Jun Ōhara defined a famous Möbius energy based on electrostatic ideas. A central property of the Möbius energy is that there are infinite energy barriers to passing the knot through itself, and with some extra conditions, only finitely many knot types can have energy below a given bound. Later work by Freedman, He, and Wang removed those extra conditions.
Other knot energies come from geometric ideas. Tangent-point energies (González and Maddocks) use the inverse of the radius of the circle tangent to the knot at one point and passing through another. A related approach is the Menger curvature energy, which uses the inverse radius of the circle through triples of points on the knot. These energies also measure how tightly or smoothly a knot sits in space.
This page was last edited on 2 February 2026, at 07:46 (CET).