Braid Theory - Closed Braids

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. A theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the "closure" of a braid. Compare with string links.

Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. Markov (1935) describes two moves on braid diagrams that yield equivalence in the corresponding closed braids. A single-move version of Markov's theorem, was published by Lambropoulou & Rourke (1997).

Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid.