Braid Group

Also called Artin Braid Groups. Consider $n$ 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, $n$ is called the Braid Index. Now enumerate the possible braids in a group, denoted $B_n$. A general $n$-braid is constructed by iteratively applying the $\sigma_i$ ( $i=1,\ldots,n-1$) operator, which switches the lower endpoints of the $i$th and $(i+1)$th strings--keeping the upper endpoints fixed--with the $(i+1)$th string brought above the $i$th string. If the $(i+1)$th string passes below the $i$th string, it is denoted $\sigma^{-1}_i$.

Topological equivalence for different representations of a Braid Word $\prod_i \sigma_i$ and $\prod_i \sigma_i'$ is guaranteed by the conditions

$$\cases{ \sigma_i\sigma_j = \sigma_j\sigma_i & for $\vert i-... ...}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} & for all $i$\cr}$$

as first proved by E. Artin. Any $n$-braid is expressed as a Braid Word, e.g., $\sigma_1\sigma_2\sigma_3\sigma_2^{-1}\sigma_1$ is a Braid Word for the braid group $B_3$. When the opposite ends of the braids are connected by nonintersecting lines, Knots are formed which are identified by their braid group 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.

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.