Birman-Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of the Birman-Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid index. As applications, we give a solution of the braid index problem, the problem to determine the braid index of a given link, and provide estimates of the crossing number of connected sums or satellites.
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