Braid group
In mathematics, the braid group on strands (denoted ), also known as the Artin braid group, is the group whose elements are equivalence classes of -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). 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 ); and in monodromy invariants of algebraic geometry. Introduction In this introduction let ; the generalization to other values of 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: {| valign="centre" |----- | | is different from |} On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered the same braid: {| valign="centre" |----- | | is the same as |} All strands are required to move from left to right; knots like the following are not considered braids: {| valign="centre" |----- | is not a braid |} 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: {| valign="centre" |----- | | composed with | | yields |} Another example: The composition of the braids and is written as . The set of all braids on four strands is denoted by . 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.) Applications Braid theory has recently been applied to fluid mechanics, specifically to the field of chaotic mixing in fluid flows. The braiding of (2 + 1)-dimensional space-time trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almost-invariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen–Thurston classification. Another field of intense investigation involving braid groups and related topological concepts in the context of quantum physics is in the theory and (conjectured) experimental implementation of so-called anyons. These may well end up forming the