Braid Group


Consider n strings, each oriented vertically from a lower to an upper "bar." If this is the least number of strings needed to make a closed braid representation of a link, n is called the braid index. A general n-braid is constructed by iteratively applying the sigma_i (i=1,...,n-1) operator, which switches the lower endpoints of the ith and (i+1)th strings--keeping the upper endpoints fixed--with the ith string brought above the (i+1)th string. If the ith string passes below the (i+1)th string, it is denoted sigma_i^(-1).

The operations sigma_i and sigma_i^(-1) on n strings define a group known as the braid group or Artin braid group, denoted B_n.

Topological equivalence for different representations of a braid word product_(i)sigma_i and product_(i)sigma_i^' is guaranteed by the conditions

 {sigma_isigma_j=sigma_jsigma_i   for |i-j|>=2; sigma_isigma_(i+1)sigma_i=sigma_(i+1)sigma_isigma_(i+1)   for all i

as first proved by E. Artin.

Any n-braid can be expressed as a braid word, e.g., sigma_1sigma_2sigma_3sigma_2^(-1)sigma_1 is a braid word in the braid group B_4. When the opposite ends of the braids are connected by nonintersecting lines, knots (or links) may formed that can be labeled by their corresponding braid word. The Burau representation gives a matrix representation of the braid groups.

See also

Braid, Braid Index, Braid Word, Knot, Link

Explore with Wolfram|Alpha


Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 132-133, 1994.Birman, J. S. "Braids, Links, and the Mapping Class Groups." Ann. Math. Studies, No. 82. Princeton, NJ: Princeton University Press, 1976.Birman, J. S. "Recent Developments in Braid and Link Theory." Math. Intell. 13, 52-60, 1991. Christy, J. "Braids.", V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335-388, 1987.Murasugi, K. and Kurpita, B. I. A Study of Braids. Dordrecht, Netherlands: Kluwer, 1999.

Referenced on Wolfram|Alpha

Braid Group

Cite this as:

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

Subject classifications