  The conformal module of conjugacy classes of braids is an invariant that
appeared earlier than the entropy of conjugacy classes of braids, and is
inverse proportional to the entropy. Using the relation between the two
invariants we give a short conceptional proof of an earlier result on the
conformal module. Mainly, we consider situations, when the conformal module of
conjugacy classes of braids serves as obstruction for the existence of
homotopies (or isotopies) of smooth objects involving braids to the respective
holomorphic objects, and present theorems on the restricted validity of
Gromov's Oka principle in these situations.
