Braid Group
Also called Artin Braid Groups. Consider 
strings, each oriented vertically from a lower to anupper ``bar.'' If this is the least number of strings needed to make a closed braid representation of a Link, iscalled the Braid Index. Now enumerate the possible braids in a group, denoted 
. A general -braid isconstructed by iteratively applying the 
(
) operator, which switches the lower endpoints of the 
thand 
th strings--keeping the upper endpoints fixed--with the th string brought above the th string.If the th string passes below the th string, it is denoted 
.
Topological equivalence for different representations of a Braid Word 
and
is guaranteed by the conditions
as first proved by E. Artin. Any
-braid is expressed as a Braid Word, e.g.,
is a Braid Word for the braid group 
. When the opposite endsof the braids are connected by nonintersecting lines, Knots are formed which are identified by their braidgroup and Braid Word. The Burau Representation gives a matrix representation of the braid groups.
References
Birman, J. S. ``Braids, Links, and the Mapping Class Groups.'' Ann. Math. Studies, No. 82. Princeton, NJ: Princeton University Press, 1976. Birman, J. S. ``Recent Developments in Braid and Link Theory.'' Math. Intell. 13, 52-60, 1991. Jones, V. F. R. ``Hecke Algebra Representations of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.
Christy, J. ``Braids.''http://www.mathsource.com/cgi-bin/MathSource/Applications/Mathematics/0202-228. Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.
- Dual Basis
- Dual Bivector
- Dual Graph
- Duality Principle
- Duality Theorem
- Dual Map
- Dual Polyhedron
- Dual Scalar
- Dual Solid
- Dual Tensor
- Dual Voting
- Du Bois Raymond Constants
- Duffing Differential Equation
- Duhamel's Convolution Principle
- Dumbbell Curve
- Duodecillion
- Dupin's Cyclide
- Dupin's Indicatrix
- Dupin's Theorem
- Duplication Formula
- Duplication of the Cube
- Durand's Rule
- Durfee Polynomial
- Durfee Square
- Dvoretzky's Theorem