On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon
Résumé
Motivated by the question of the polytopal realizability of the simplicial complex $\Gamma_{n,k}$ of $(k+1)$-crossing-free sets of diagonals of the convex $n$-gon, we study the first open case, namely when $n=8$ and $k=2$. We give a complete description of the space of symmetric realizations of $\Gamma_{8,2}$, that is, of the polytopes $P$ whose boundary complex is isomorphic to $\Gamma_{8,2}$, and such that the natural action of the dihedral group on $\Gamma_{8,2}$ defines an action on $P$ by isometry.