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.
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 
.
The operations 
and 
on 
strings define a group known as
the braid group or Artin braid group, 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."