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 ,

(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.

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.