Let Dn denote the n-punctured disk in the complex plane, where the punctures are on the real axis. An n-braid α is said to be reducible if there exists an essential curve system C in Dn, called a reduction system of α, such that α ∗ C = C where α ∗ C denotes the action of the braid α on the curve system C. A curve system C in Dn is said to be standard if each of its components is isotopic to a ...