Bounds for the crossing number in terms of the braid index?

Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?

For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for braid index 3 is 4, the smallest crossing number for braid index 4 is 6, the smallest crossing number for braid index 5 is 8, the smallest crossing number for braid index 6 is 10, and the smallest crossing number for braid index 7 is 12. The urge to extrapolate is strong.

There is no upper bound, of course -- already with braid index 2 there are knots with arbitrarily large (odd) crossing number.