  Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $\Phi_{\rm KZ}$
by considering the regularized holonomy of the KZ connection along the {\em
droit chemin} $[0,1]$. The KZ associator is a group-like element of the free
associative algebra with two generators, and it satisfies the pentagon
equation.
  In this paper, we consider paths on $\mathbb{C}\backslash \{ z_1, \dots,
z_n\}$ which start and end at tangential base points. These paths are not
necessarily straight, and they may have a finite number of transversal
self-intersections. We show that the regularized holonomy $H$ of the KZ
connection associated to such a path satisfies a generalization of Drinfeld's
pentagon equation. In this equation, we encounter $H$, $\Phi_{\rm KZ}$, and new
factors associated to self-intersections, to tangential base points, and to the
rotation number of the path.
