Braid Index

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 $c(L)$ and braid index $i(L)$,

$$c(L)\geq 2[i(L)-1]$$

(Ohyama 1993). 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 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.