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 stringskeeping the upper endpoints fixedwith 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.