Abstract Brick polytopes constitute a remarkable family of polytopes associated to the spherical subword complexes of Knutson and Miller. They were introduced for finite Coxeter groups by Pilaud and Stump, who used them to produce geometric realizations of generalized associahedra arising from the theory of cluster algebras of finite types. In this paper, we present an application of the vast generalization of brick polyhedra for general subword complexes (not necessarily spherical) recently introduced by Jahn and Stump. More precisely, we show that the $$\nu $$ ν -associahedron, a polytopal complex whose edge graph is the Hasse diagram of the $$\nu $$ ν -Tamari lattice introduced by Préville-Ratelle and Viennot, can be geometrically realized as the complex of bounded faces of the brick polyhedron of a well chosen subword complex. We also present a suitable projection to the appropriate dimension, which leads to an elegant vertex-coordinate description.

A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani (Aequationes Math 34(2-3):287-297, 1987, 10.1007/BF01830678), via a non-constructive proof using topological tools from homology theory. An elegant constructive proof was given by Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai (2009, https://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/). In this paper, we show that Kalai's conjecture holds in the particular case of Knutson and Miller's spherical subword complexes. This family of simplicial spheres arises in the context of Coxeter groups, and is conjectured to be polytopal. In contrast, not all manifolds are reconstructible. We show two explicit examples, namely the torus and the projective plane.

Abstract. This is the first contribution of a sequence of papers introducing the notions of [Formula: see text]-weak order and [Formula: see text]-permutahedra, certain discrete objects that are indexed by a sequence of nonnegative integers [Formula: see text]. In this first paper, we concentrate purely on the combinatorics and lattice structure of the [Formula: see text]-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the [Formula: see text]-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the [Formula: see text]-weak order to certain trees gives rise to the [Formula: see text]-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the [Formula: see text]-Tamari lattice can be obtained as a quotient lattice of the [Formula: see text]-weak order when [Formula: see text] has no zeros, and show that the [Formula: see text]-Tamari lattices (for arbitrary [Formula: see text]) are isomorphic to the [Formula: see text]-Tamari lattices of Préville-Ratelle and Viennot. The underlying geometric structure of the [Formula: see text]-weak order will be studied in a sequel of this paper, where we introduce the notions of [Formula: see text]-permutahedra and [Formula: see text]-associahedra.

We study the motion of a robotic arm inside a rectangular tunnel. We prove that the configuration space of all possible positions of the robot is a ${CAT}(0)$ cubical complex. This allows us to use techniques from geometric group theory to find the optimal way of moving the arm from one position to another. We also compute the diameter of the configuration space, that is, the longest distance between two positions of the robot.