Syntactic aspects of hypergraph polytopes

Abstract

This paper introduces an inductively defined tree notation for allthe faces of polytopes arising from a simplex by truncations. This notation al-lows us to view inclusion of faces as the process of contracting tree edges. Ournotation instantiates to the well-known notations for the faces of associahedraand permutohedra. Various authors have independently introduced combina-torial tools for describing such polytopes. We build on the particular approachdeveloped by Doˇsen and Petri ́c, who used the formalism of hypergraphs to de-scribe the interval of polytopes from the simplex to the permutohedron. Thisinterval was further stretched by Petric to allow truncations of faces that arethemselves obtained by truncations, and iteratively so. Our notation applies toall these polytopes. We illustrate this by showing that it instantiates to a no-tation for the faces of the permutohedron-based associahedra, that consists ofparenthesised words with holes. Dosen and Petric have exhibited some familiesof hypergraph polytopes (associahedra, permutohedra, and hemiassociahedra)describing the coherences, and the coherences between coherences etc., arisingby weakening sequential and parallel associativity of operadic composition. Wecomplement their work with a criterion allowing us to recover the informationwhether edges of these “operadic polytopes” come from sequential, or fromparallel associativity. We also give alternative proofs for some of the originalresults of Dosen and Petric.