A braid is an intertwining of some number of strings attached to top and bottom "bars" such that each string never "turns back up." In other words, the path of each string in a braid could be traced out by a falling object if acted upon only by gravity and horizontal forces. A given braid may be assigned a symbol known as a braid word that uniquely identifies it (although equivalent braids may have more than one possible representations). For example, sigma_1sigma_3sigma_1sigma_4^(-1)sigma_2sigma_4^(-1)sigma_2sigma_4^(-1)sigma_3sigma_2^(-1)sigma_4^(-1) is a braid word for the braid illustrated above.

If K is a knot and


where Delta_K(x) is the Alexander polynomial of K, then K cannot be represented as a closed 3-braid. Also, if


then K cannot be represented as a closed 4-braid (Jones 1985).

See also

Braid Group, Braid Index, Braid Word, Knot, Link

Explore with Wolfram|Alpha


Artin, E. "The Theory of Braids." Amer. Sci. 38, 112-119, 1950. Christy, J. "Braids.", V. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 1985.Murasugi, K. and Kurpita, B. I. A Study of Braids. Dordrecht, Netherlands: Kluwer, 1999.

Referenced on Wolfram|Alpha


Cite this as:

Weisstein, Eric W. "Braid." From MathWorld--A Wolfram Web Resource.

Subject classifications