Knot energy
Encyclopedia
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 C2 functions. A property of the functional often requires that evolution of the knot under gradient descent
does not change knot type.
The most common type of knot energy comes from the intuition of the knot as electrically charged. Coulomb's law
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
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-intersections are prevented.
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
. 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—Wang removed these restrictions.
Physical knot theory
Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by physical considerations from biology, chemistry, and physics. Traditional knot theory models a knot as a simple closed loop in three dimensional space. Such a knot has no thickness or physical...
, a knot energy is a functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
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
Well-behaved
Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...
class. For example, one may consider only polygonal circles or C2 functions. A property of the functional often requires that evolution of the knot under gradient descent
Gradient descent
Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point...
does not change knot type.
The most common type of knot energy comes from the intuition of the knot as electrically charged. Coulomb's law
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
states that two electric charges of the same sign will repel each other as the inverse square of the distance
Inverse-square law
In physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity....
. Thus the knot will evolve under gradient descent according to the electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
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-intersections are prevented.
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
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...
. 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—Wang removed these restrictions.