Braid Index
المؤلف:
Franks, J. and Williams, R. F.
المصدر:
"Braids and the Jones Polynomial." Trans. Amer. Math. Soc. 303
الجزء والصفحة:
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7-6-2021
4449
Braid Index
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.
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.
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