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We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators ${}_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators ${\Delta }_{ij}^k$ as $log\left(\right|k|+1)$ instead of simply $\left|k\right|$. The geometrical complexity is some natural measure of the amount of distortion of the $n$ times punctured disk caused by a homeomorphism. Our main...