Deformations of associahedra and visibility graphs

Given an arbitrary simple polygon, we construct a polytopal complex analogous to the associahedron based on its convex diagonalizations. This polytopal complex is shown to be contractible, and a geometric realization is provided based on the theory of secondary polytopes. We then reformulate a combinatorial deformation theory in terms of visibility and presents some open problems

Hypersimplicial subdivisions

Let π:Rn→Rd be any linear projection, let A be the image of the standard basis. Motivated by Postnikov’s study of postitive Grassmannians via plabic graphs and Galashin’s connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the poset of subdivisions induced by the restriction of π to the k-th hypersimplex, for k=1,…,n−1 . We show that: For arbitrary A and for k≤d+1 , the corresponding fiber polytope F(k)(A) is normally isomorphic to the Minkowski sum of the secondary polytopes of all subsets of A of size max{d+2,n−k+1} . When A=Pn is the vertex set of an n-gon, we answer the Baues question in the positive: the inclusion of the poset of π -coherent subdivisions into the poset of all π -induced subdivisions is a homotopy equivalence. When A=C(d,n) is the vertex set of a cyclic d-polytope with d odd and any n≥d+3, there are non-lifting (and even more so, non-separated) π -induced subdivisions for k=2.The authors were supported by the Einstein Foundation Berlin under grant EVF-2015-230. Work of F. Santos is also supported by grants MTM2017-83750-P/AEI/10.13039/501100011033 and PID2019-106188GB-I00/AEI/10.13039/501100011033 of the Spanish State Research Agency

Algorithms for distance problems in planar complexes of global nonpositive curvature

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex K with n vertices, one can construct in O(n^2 log n) time a data structure D of size O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x and the query point y can be computed in linear time. Our second algorithm computes the convex hull of a finite set of points in a CAT(0) planar complex. This algorithm is based on Toussaint's algorithm for computing the convex hull of a finite set of points in a simple polygon and it constructs the convex hull of a set of k points in O(n^2 log n + nk log k) time, using a data structure of size O(n^2 + k)

Celebrating Loday’s associahedron

We survey Jean-Louis Loday’s vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry, and algebra. We present in particular four topics where it plays a central role: lattice congruences of the weak order and their quotientopes, cluster algebras and their generalized associahedra, nested complexes and their nestohedra, and operads and the associahedron diagonal

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum