Realizations of the associahedron and cyclohedron

Abstract

We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott-Taubes polytope) and compare them to the permutahedron of type A_n and B_n respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type A_n or B_n respectively as only input and which specialises to a procedure presented by J.-L. Loday for a certain orientation of A_n. The described realizations have cambrian fans of type A and B as normal fans. This settles a conjecture of N. Reading for cambrian fans of these types.Comment: v2: 18 pages, 7 figures; updated version has revised introduction and updated Section 4; v3: 21 pages, 2 new figures, added statement (b) in Proposition 1.4. and 1.7 plus extended proof; added references [1], [29], [30]; minor changes with respect to presentatio