A quantitative Birman–Menasco finiteness theorem and its application to crossing number

Abstract

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.