Symmetric edge polytopes are a recent and well-studied family of centrally symmetric polytopes arising from graphs. In this paper, we introduce a generalization of this family to arbitrary simplicial complexes. We show how topological properties of a simplicial complex can be translated into geometric properties of such polytopes, and vice versa. We study the integer decomposition property, facets and reflexivity of these polytopes. Using Gröbner basis techniques, we obtain a (not necessarily unimodular) triangulation of these polytopes. Due to the tools we use, most of our results hold in the more general setting of arbitrary centrally symmetric polytopes.
When modeling a combinatorial object based on an underlying graph G, it is often the case that the incidence matrix of G is a valuable tool for understanding the corresponding object. A prime example of this is the sandpile group [26,Chapter 4] of a graph G, where the size of the group can be computed via the incidence matrix using the well-known Matrix Tree theorem [2].
From a topological point of view, interpreting a graph G as a 1-dimensional simplicial complex, the incidence matrix of G is just its first boundary map in homology. Having this connection in mind, in 2010 Duval, Klivans and Martin [13] introduced the (co)critical groups of an arbitrary simplicial complex ∆, with the goal of generalizing the Matrix Tree theorem to higher dimensional simplicial complexes.
From a different perspective, another object defined in terms of the incidence matrix of a graph G that has received a lot of attention recently is the so-called symmetric edge polytope P G (see e.g., [5,6,7,22,24,34]) of G, defined as
where ∂ 1 denotes the incidence matrix of G. Symmetric edge polytopes are of importance in various areas of Mathematics, as e.g., in the study of finite metric spaces and computational phylogenetics, and also in Physics through their connection to the Kuramoto model for interacting oscillators (see [7] for more details). Many combinatorial properties of symmetric edge polytopes have been shown to not depend on the underlying graph explicitly but just on the graphic matroid it represents. Motivated by this, symmetric edge polytopes have been generalized to regular matroids in [8] (see also [9] for more results on these).
In this paper, we aim at a more topological generalization of symmetric edge polytopes. Namely, interpreting a graph and its incidence matrix as a 1-dimensional simplicial complex and its first boundary map in homology, respectively, (see also [13]), generalize symmetric edge polytopes to arbitrary simplicial complexes as follows: Given a d-dimensional simplicial complex ∆ with top boundary map (in homology) ∂ d , we call (1)
the symmetric homology polytope and the symmetric cohomology polytope of ∆, respectively, where, for a matrix A, the notation conv[A| -A] means that we take the convex hull over the columns of A and -A. The goal of this paper is to initiate the study of these two new classes of polytopes and to motivate further research in them. We will see that sometimes they behave very similar to their classical counterparts, but sometimes they also show a different behavior. More precisely, after having computed basic properties, as the dimension and their number of vertices, our first main result is the precise description of the faces of a symmetric homology polytope. From this, we, in particular, derive a characterization of their facets via labelings of the facets of ∆ and spanning forests (see Theorem 3.6). This result generalizes the facet description of the symmetric edge polytope of a graph in a very natural way (see [22,Theorem 3.1]). Moreover, Theorem 3.6 allows us to compute the number of facets of P ∆ explicitly if the underlying simplicial complex is a connected closed orientable pseudomanifold (see Corollary 3.9). In the 1-dimensional situation, the obtained formula specializes to the known formula of the number of facets of the symmetric edge polytope of a cycle and hence provides another parallel to symmetric edge polytopes of graphs (cf., [7,Proposition 4.3]).
Interestingly, another similarity, which is a dissimilarity at the same time, is given by the Gröbner basis of the toric ideal of these polytopes: In Theorem 4.1, we manage to compute a Gröbner basis for the toric ideals of any centrally symmetric lattice polytope. On the one hand, compared to the Gröbner basis from [22,Proposition 3.8], our Gröbner basis only requires one completely new type of binomials and thus is a nice generalization. On the other hand, while toric ideals of symmetric edge polytopes of graphs admit a squarefree Gröbner basis, this is no longer true for symmetric (co)homology polytopes (see Theorem 4.4). Consequently, symmetric (co)homology polytopes do not necessarily have the integer decomposition property anymore and we give a necessary criterion for this property to hold in Proposition 5.3. In particular, we show that torsion in the (d -1) st homology group of a d-dimensional simplicial complex is an obstruction for both types of polytopes. This is one example where we are able to translate topological properties of ∆ into geometric properties of the polytopes P ∆ and P ∆ .
The main source for the different behaviors of symmetric (co)homology polytopes compared to their more classical counterparts for graphs lies in the different natures of the boundary matrices. Indeed, while incidence matrices of graphs are always totally unimodular, this is not true for arbitrary boundary matrices. For symmetric edge polytopes, totally unimodularity was crucial in the proof of various propertie
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