Braid Index
The least number of strings needed to make a closed braid representation of a Link. The braid index is equal tothe 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 anoriented 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 theexceptions 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.
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