In
physical knot theory, a knot energy is a
functional on the space of all knot conformations. A conformation of a knot is a particular embedding of a circle into three-dimensional space. Depending on the needs of the energy function, the space of conformations is restricted to a sufficiently
nicely behaved class. For example, one may consider only polygonal circles or ''C''
2 functions. A property of the functional often requires that evolution of the knot under
gradient descent
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function.
The idea is to take repeated steps in the opposite direction of the gradi ...
does not change knot type.
Electrical charge
The most common type of knot energy comes from the intuition of the knot as
electrically charged
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
.
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
states that two electric charges of the same sign will repel each other as the
inverse square of the distance. Thus the knot will evolve under gradient descent according to the
electric potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
to an ideal configuration that minimizes the electrostatic energy. Naively defined, the integral for the energy will diverge and a regularization trick from physics, subtracting off a term from the energy, is necessary. In addition the knot could change knot type under evolution unless
self-intersection
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
s are prevented.
Variations
An electrostatic energy of polygonal knots was studied by Fukuhara in 1987 and shortly after a different, geometric energy was studied by Sakuma. In 1988,
Jun O'Hara defined a knot energy based on electrostatic energy,
Möbius energy
In mathematics, the Möbius energy of a knot is a particular knot energy, i.e., a functional on the space of knots. It was discovered by Jun O'Hara, who demonstrated that the energy blows up as the knot's strands get close to one another. This ...
. A fundamental property of the O'Hara energy function is that infinite energy barriers exist for passing the knot through itself. With some additional restrictions, O'Hara showed there were only finitely many knot types with energies less than a given bound. Later, Freedman, He, and Wang removed these restrictions.
[.]
References
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Knot theory