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.