Also called Artin Braid Groups. Consider strings, each oriented vertically from a lower to an upper ``bar.'' If this is the least number of strings needed to make a closed braid representation of a Link, is called the Braid Index. Now enumerate the possible braids in a group, denoted . A general -braid is constructed by iteratively applying the ( ) operator, which switches the lower endpoints of the th and 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
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.
Christy, J. ``Braids.''
http://www.mathsource.com/cgi-bin/MathSource/Applications/Mathematics/0202-228.
Jones, V. F. R. ``Hecke Algebra Representations of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.
Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.