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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. 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 Group s of ''Y'' are trivial. 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.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. Alexander 's Theorem observes 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 . Markov 's Theorem describes two moves on braid diagrams which yield equivalence in the corresponding closed braids. A single-move version of Markov's theorem, due to Sofia Lambropoulou and Colin Rourke, was published in 1997. 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 REFERENCES
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