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 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.

© 1996-9

1999-05-26