  We define invariants of braids rather than invariants of conjugacy classes of
braids. For any pure three-braid we give effective upper and lower bounds for
these invariants. This is done in terms of a natural syllable decomposition of
the word representing the image of the braid in the braid group modulo its
center. The bounds differ by a multiplicative constant not depending on the
word. Respective bounds are given for all three-braids. We also obtain
effective upper and lower bounds for the entropy of pure three-braids in these
terms. The proof leads to the study of the extremal length of classes of curves
representing elements of the fundamental group of the twice punctured complex
plane.
