Braid Word

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). In particular, an -braid can constructed by iteratively applying the  () operator, which switches the lower endpoints of the th and th strings--keeping the upper endpoints fixed--with the th string brought above the th string. If the th string passes below the th string, it is denoted .

An ordered combination of the  and  symbols constitutes a braid word. For example,  is a braid word for the braid illustrated above, where the symbols can be read off the diagram left to right and then top to bottom.

By Alexander's theorem, any link is representable by a closed braid, but there is no general procedure for reducing a braid word to its simplest form. However, Markov's theorem gives a procedure for identifying different braid words which represent the same link.

The following table lists (not necessarily unique) braid words for some common knots and links.

link	braid word
Borromean rings
figure eight knot
Hopf link
Miller Institute knot
Solomon's seal knot
stevedore's knot
trefoil knot
Whitehead link

Let  be the sum of positive exponents, and  the sum of negative exponents in the braid group . If

then the closed braid  is not amphichiral (Jones 1985).

REFERENCES:

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 132-133, 1994.

Jones, V. F. R. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 1985.

Jones, 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.