Zigzag structure of complexes

Inspired by Coxeter's notion of Petrie polygon for $d$-polytopes (see \cite{Cox73}), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of $d$-polytopes, including semiregular, regular-faced, Wythoff Archimedean ones, Conway's 4-polytopes, half-cubes, folded cubes. Also considered are regular maps and Lins triality relations on maps.