Braid group
In mathematics, the braid group on n strands (denoted Bn), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (eg. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.
In this introduction let n = 4; the generalization to other values of n will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected with an item of the second set so that a one-to-one correspondence results. Such a connection is called a braid. Often some strands will have to pass over or under others, and this is crucial: the following two connections are different braids:
On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered the same braid:
All strands are required to move from left to right; knots like the following are not considered braids:
Any two braids can be composed by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:
Another example:
The composition of the braids σ and τ is written as στ.
The set of all braids on four strands is denoted by B4. The above composition of braids is indeed a group operation. The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its centre. (The first two example braids above are inverses of each other.)
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