A geometric braid with n = 3 strings. This particular example is a
standard pigtail braid. Also shown is the corresponding Borromean link.
Recent Braid Theory Papers and Reviews
Click on the summary links below to see
abstracts and illustrations.
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
Energy-crossing number relations for braided magnetic fields
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
Abstract
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
Abstract
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
Abstract
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.
