The braid is associated with a planar graph
The 24 elements of a Permutation group on 4 elements as braids. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation Note that all crossings shown are of the left-over-right sort and other choices are possible. Indeed, the braid group on four strands is infinite.

In topology, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalisations. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. A braid (also called plait) is a complex structure or pattern formed by intertwining three or more strands of flexible material such as textile fibers wire or human hair The idea is that braids can be organised into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Such groups may be described by explicit presentations, as was shown by Emil Artin. In Mathematics, one method of defining a group is by a presentation. Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician For an elementary treatment along these lines, see the article on braid groups. In Mathematics, the braid group on n strands, denoted by B n, is a certain group which has an intuitive geometrical representation Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. "Configuration space" may also refer to PCI Configuration Space.

To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold X of dimension at least 2. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The symmetric product of n copies of X means the quotient of the n-fold Cartesian product Xn of X with itself, by the permutation action of the symmetric group on n letters operating on the indices of coordinates. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying That is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

A path in the n-fold symmetric product is the abstract way of discussing n points of X, considered as an unordered n-tuple, independently tracing out n strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace Y of the symmetric product, of orbits of n-tuples of distinct points. That is, we remove all the subspaces of Xn defined by conditions xi = xj. This is invariant under the symmetric group, and Y is the quotient by the symmetric group of the non-excluded n-tuples. Under the dimension condition Y will be connected.

With this definition, then, we can call the braid group of X with n strings the fundamental group of Y (for any choice of base point - this is well-defined up to isomorphism). In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose The case of X the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher homotopy groups of Y are trivial. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional

## Closed braids

When X is the plane, the braid can be closed, i. e. , corresponding ends can be connected in pairs, to form a link, i. In Mathematics, a link is a collection of knots which do not intersect but which may be linked (or knotted together e. , a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, depending on the permutation of strands determined by the link. A theorem of J. W. Alexander demonstrates that every link can be obtained in this way from a braid. James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of

Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces Markov's Theorem describes two moves on braid diagrams which yield equivalence in the corresponding closed braids. Andrey (Andrei Andreyevich Markov (Андрей Андреевич Марков (June 14 1856 N A single-move version of Markov's theorem, due to Sofia Lambropoulou and Colin Rourke, was published in 1997. Year 1997 ( MCMXCVII) was a Common year starting on Wednesday (link will display full 1997 Gregorian calendar

Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid. Vaughan Frederick Randal Jones DCNZM (born 31 December 1952) is a New Zealand Mathematician, known for his work on In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983