Energy-crossing number relations for braided magnetic fields
1993 Physical Review Letters 70 705-708
Minimum crossing numbers for three - braids
1994 Journal of Physics A: Mathematical and General 27 6205-6213
Hamiltonian dynamics generated by Vassiliev invariants
2001 Journal of Physics A: Mathematical and General 34 1363-1374
Topological Invariants in braid theory
2001 Letters in Mathematical Physics
Two topologically equivalent braids. The braid on the right corresponds to a standard pigtail. These braids can be employed in models of topologically complex magnetic fields, such as the fields inside x-ray loops in the solar corona and in accretion disk coronae. Relaxing a braid (using computational group theory) to its simplest form corresponds to relaxing a magnetic field to its minimum (equilibrium) energy state. A close correspondence exists between braid complexity and magnetic energy storage . This correspondence allows the estimation of active region heating in the solar corona due to the dissipation of topological structure.
Minimum crossing numbers for three – braids PDF File - 612 KB
Given a braid on N strings, find an algorithm which generates an Artin braid word B of minimal length. This is an important unsolved problem -- a solution would give us the most economical way of notating and drawing braids. The length of an Artin word equals the number of crossings seen in a braid diagram. Minimum crossing numbers provide a measure of complexity for braids. This paper presents an algorithm for N = 3. Also a three dimensional configuration space for 3-braids will be defined and analyzed.
The braid Bmin is obtained from the equivalent braid B0 by use of the algorithm presented in this paper.
Hamiltonian dynamics generated by Vassiliev invariants PDF File – 410 KB
This paper employs higher order winding numbers to generate Hamiltonian motion of particles in two dimensions. The ordinary winding number counts how many times two particles rotate about each other. Higher order winding numbers measure braiding motions of three or more particles. These winding numbers relate to various invariants known in topology and knot theory, for example Massey and Milnor numbers, and can be derived from Vassiliev-Kontsevich integrals. The invariants can be regarded as complex-valued functions of the paths of the particles. The real part gives the winding number, whereas the imaginary part seems uninteresting.
In this paper, we set the imaginary part to be a Hamiltonian for particle motions. For just two particles, this gives the familiar motion of two point-vortices. However, for three or more particles, the Hamiltonian generates more complicated intertwining patterns. We examine the dynamics for the case of 3 particles, and show that the motion is completely integrable. The intertwining patterns correspond to periodic braids; closure of these braids gives links such as the Borromean rings. The Hamiltonian provides an elegant method for generating simple geometrical examples of complicated braids and links.
Topological Invariants in braid theory PDF File – 761 KB
Many invariants of knots and links have their counterparts in braid theory. Often these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is winding number -- the net number of times two strings in a braid wrap about each other. But other, higher order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher order invariants using only elementary differential geometry.
Some of the higher order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.
Some of the higher order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.