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. 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.
braid | braid group | braid word | knot | link
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