Morton-Franks-Williams Inequality

Let $E$ be the largest and $e$ the smallest Power of $\ell$ in the HOMFLY Polynomial of an oriented Link, and $i$ be the Braid Index. Then the Morton-Franks-Williams Inequality holds,

$$i\geq {\textstyle{1\over 2}}(E-e)+1$$

(Franks and Williams 1985, Morton 1985). 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.

See also Braid Index

References

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