Wigglyhedra

TL;DR

This paper introduces the wiggly complex and wigglyhedron, connecting combinatorial, geometric, and categorical structures, and explores their properties, bijections, and relations to known polytopes like Cambrian associahedra.

Contribution

It defines the wiggly complex and wigglyhedron, establishes bijections with wiggly permutations, and relates these structures to categorical objects and known polytopes.

Findings

Wiggly pseudotriangulations form the facets of a $(2n-1)$-dimensional pseudomanifold.

Wiggly permutations avoid specific patterns and define the wiggly lattice.

The wiggly complex is isomorphic to the boundary of the polar of the wigglyhedron.

Abstract

Motivated by categorical representation theory, we define the wiggly complex, whose vertices are arcs wiggling around $n+2$ points on a line, and whose faces are sets of wiggly arcs which are pairwise pointed and non-crossing. The wiggly complex is a $(2n-1)$-dimensional pseudomanifold, whose facets are wiggly pseudotriangulations. We show that wiggly pseudotriangulations are in bijection with wiggly permutations, which are permutations of $[2n]$ avoiding the patterns $(2j-1) \cdots i \cdots (2j)$ for $i < 2j-1$ and $(2j) \cdots k \cdots (2j-1)$ for $k > 2j$. These permutations define the wiggly lattice, an induced sublattice of the weak order. We then prove that the wiggly complex is isomorphic to the boundary complex of the polar of the wigglyhedron, for which we give explicit and simple vertex and facet descriptions. Interestingly, we observe that any Cambrian associahedron is normally equivalent to a well-chosen face of the wigglyhedron. Finally, we recall the correspondence of wiggly arcs with objects in a category, and we develop categorical criteria for a subset of wiggly arcs to form a face of the wiggly complex.