Discrete Comput Geom 37:517–543 (2007) Discrete & Computational DOI: 10.1007/s00454-007-1319-6 Geometry © 2007 Springer Science+Business Media, Inc. Realizations of the Associahedron and Cyclohedron∗ Christophe Hohlweg1 and Carsten E. M. C. Lange2 1The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1 chohlweg@fields.utoronto.ca 2Fachbereich f¨ur Mathematik und Informatik, Freie Universit¨at Berlin, Arnimallee 3, D-14195 Berlin, Germany
[email protected] Abstract. We describe many different realizations with integer coordinates for the associ- ahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott–Taubes polytope) and compare them with the permutahedron of type A and B, respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type An or Bn as the only input data and which specializes to a procedure presented by J.-L. Loday for a certain orientation of An. The described realizations have cambrian fans of type A and B as normal fans. This settles a conjecture of N. Reading for cambrian lattices of these types. 1. Introduction The associahedron Asso(An−1) was discovered by Stasheff in 1963 [27], and is of great importance in algebraic topology. It is a simple (n − 1)-dimensional convex polytope whose 1-skeleton is isomorphic to the undirected Hasse diagram of the Tamari lattice on the set Tn+2 of triangulations of an (n+2)-gon (see for instance [16]) and therefore a fun- damental example of a secondary polytope as described in [13]. Numerous realizations of the associahedron have been given, see [6], [17], and the references therein.