The braid group 
, also called the Artin braid
group, is the group generated by the braid operations 
and 
on 
strings. The strings are 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.
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
 |
(1)
|
as first proved by E. Artin.
Any 
-braid can be expressed as a braid
word, e.g., 
is a braid word in the braid group 
.
When the opposite ends of the braids are connected by nonintersecting lines, knots
(or links) may formed that can be labeled by their corresponding
braid word. The Burau
representation gives a matrix representation of the
braid groups.
Christy, J. "Braids."