Braid Index

A braid index is the least number of strings needed to make a closed braid representation of a link. The braid index is equal to the least number of Seifert circles in any projection of a knot (Yamada 1987). Also, for a nonsplittable link with link crossing number  and braid index ,

(Ohyama 1993). Let  be the largest and  the smallest power of  in the HOMFLY polynomial of an oriented link, and  be the braid index. Then the morton-franks-williams inequality holds,

(Franks and Williams 1987). The inequality is sharp for all prime knots up to 10 crossings with the exceptions of 09-042, 09-049, 10-132, 10-150, and 10-156.

REFERENCES:

Franks, J. and Williams, R. F. "Braids and the Jones Polynomial." Trans. Amer. Math. Soc. 303, 97-108, 1987.

Jones, V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335-388, 1987.

Ohyama, Y. "On the Minimal Crossing Number and the Brad Index of Links." Canad. J. Math. 45, 117-131, 1993.

Yamada, S. "The Minimal Number of Seifert Circles Equals the Braid Index of a Link." Invent. Math. 89, 347-356, 1987.