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

c(L)>=2[i(L)-1]

(Ohyama 1993). 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

(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, Braid Group, Braid Word, Knot, Link

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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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Braid Index

Cite this as:

Weisstein, Eric W. "Braid Index." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BraidIndex.html

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