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Morton-Franks-Williams Inequality

Let E be the largest and e the smallest power of l in the HOMFLY polynomial of an oriented link, and i be the braid index. Then the Morton-Franks-Williams inequality holds,

i>=1/2(E-e)+1

(Morton 1986, 1988, 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.

See also

Braid Index

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References

Franks, J. and Williams, R. F. "Braids and the Jones Polynomial." Trans. Amer. Math. Soc. 303, 97-108, 1987.Morton, H. R. "Seifert Circles and Knot Polynomials." Math. Proc. Cambridge Philos. Soc. 99, 107-109, 1986.Morton, H. R. "Polynomials from Braids." In Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference in Mathematical Sciences on Artin's Braid Group held at the University of California, Santa Cruz, California, July 13-26, 1986 (Ed. J. S. Birman and A. Libgober). Providence, RI: Amer. Math. Soc., pp. 575-585, 1988.

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Morton-Franks-Williams Inequality

Cite this as:

Weisstein, Eric W. "Morton-Franks-Williams Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Morton-Franks-WilliamsInequality.html

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