Knot energy

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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.

Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by considerations from biology, chemistry, and physics. Physical knot theory is used to study how geometric and topological characteristics of filamentary structures, such as magnetic flux tubes, vortex filaments, polymers, DNAs, influence their physical properties and functions. It has applications in various fields of science, including topological fluid dynamics, structural complexity analysis and DNA biology.

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Electrical charge

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.

Coulombs law Fundamental physical law of electromagnetism

Coulomb's law, or Coulomb's inverse-square law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force. The quantity of electrostatic force between stationary charges is always described by Coulomb’s law. The law was first published in 1785 by French physicist Charles-Augustin de Coulomb, and was essential to the development of the theory of electromagnetism, maybe even its starting point, because it was now possible to discuss quantity of electric charge in a meaningful way.

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An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point beyond the influence of the electric field charge can be used.

Variations

An electrostatic energy of polygonal knots was studied by Fukuhara in 1987 [1] and shortly after a different, geometric energy was studied by Sakuma. [2] [3] In 1988, Jun O'Hara defined a knot energy based on electrostatic energy, Möbius energy. [4] 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. [5]

Jun O'Hara, legally named Jun Imai, is a Japanese mathematician who works on the fields of low-dimensional topology and knot theory. He is a professor at Tokyo Metropolitan University.

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 is a useful property because it prevents self-intersection and ensures the result under gradient descent is of the same knot type.

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References

  1. ↑ Fukuhara, Shinji (1988), "Energy of a knot", A fête of topology, Academic Press, Boston, MA, pp. 443–451, MR   0928412 .
  2. ↑ Sakuma, M. (1987), "Problem no. 8", in Kojima, S.; Negami, S., The collection of problems on “Low dimensional topology and related matters” (in Japanese), p. 7. As cited by Langevin & O'Hara (2005).
  3. ↑ Langevin, R.; O'Hara, J. (2005), "Conformally invariant energies of knots", Journal of the Institute of Mathematics of Jussieu, 4 (2): 219–280, arXiv: math.GT/0409396 , doi:10.1017/S1474748005000058, MR   2135138 .
  4. ↑ O'Hara, Jun (1991), "Energy of a knot", Topology, 30 (2): 241–247, doi:10.1016/0040-9383(91)90010-2, MR   1098918 .
  5. ↑ Freedman, Michael H.; He, Zheng-Xu; Wang, Zhenghan (1994), "Möbius energy of knots and unknots", Annals of Mathematics , Second Series, 139 (1): 1–50, doi:10.2307/2946626, MR   1259363 .
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