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 two. The *symmetric product* of n copies of X means the quotient of the n-fold cartesian product X^{n} of X with itself, by the permutation action of the symmetric group on n letters operating on the indices of coordinates. That is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it.

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 X^{n} defined by conditions *x*_{i} = *x*_{j}. 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). 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.

When X is the plane, the braid can be *closed*, i.e., corresponding ends can be connected in pairs, to form a link, i.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. J. W. Alexander observed that every link can be obtained in this way from a braid. Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. The Jones polynomial is defined, a priori, as a braid invariant and then shown to depend only on the class of the closed braid.

See also knot theory, braid group.