Poincaré polynomial of elliptic arrangements is not determined by the Tutte polynomial
Introduction
Let be an integer matrix and let be a group of the form , where is a finite abelian group. Each column of defines a morphism from to given by We call the kernel of the map defined by the -column of . The complement of the arrangement in is the topological space
When we obtain the definition of central hyperplane arrangements (with rational equations). If the arrangement is called toric. We are mainly interested in the case (an elliptic curve), these arrangements are called elliptic arrangement.
There are several combinatorial objects associated with an arrangement: for instance, the poset of layers, the arithmetic matroid [3], [4] and the -Tutte polynomial [7], [8], [12]. Given a subset of we call layer any connected component of the intersection . The poset of layers is the set of all layers ordered by reverse inclusion. The arithmetic matroid is the triple associated with toric, hyperplanes or elliptic arrangements, where and are, respectively, the codimension and the number of connected components of . The -Tutte polynomial is a generalization of the arithmetic Tutte polynomial and of the classical Tutte polynomial; it is defined by where is the multiplicity function defined in [7, Definition 4.6]; if is connected, then coincides with the number of connected components of .
Recently, a formula for the Poincaré polynomial of was found by Liu, Tran and Yoshinaga [7] when is not compact, i.e. . This formula involves the -characteristic polynomial , which is a specialization of the -Tutte polynomial: When is not compact, the Poincaré polynomial of is where is the Poincaré polynomial of the group . The formula for the Euler characteristic holds for all groups ( is the Euler characteristic of ), see [2], [7].
We focus on the “smallest” compact group , the case being trivial. From now on, we denote the two-dimensional compact torus by . In this case, Bibby [2] and Dupont [6] have given a model of the cohomology ring , provided by the second page of the Serre spectral sequence for the inclusion . As shown in [10], this model is combinatorial, i.e. can be defined from the arithmetic matroid . Thus the Betti numbers are implicitly encoded in the arithmetic matroid, but there is no explicit formula that allows their calculation. We will show that these Betti numbers are independent from the arithmetic Tutte polynomial, exhibiting an example.
Section snippets
The model for cohomology
We recall the model developed by Dupont [6] and Bibby [2] for the cohomology ring in the particular case of graphic elliptic arrangements.
Let be the quotient of by the diagonal action of . Given a finite graph , undirected and without loops or multiple edges, we can define an arrangement in given by the divisors for each edge . We fix arbitrarily a spanning forest of and an orientation of . Consider the external
The example
Consider the two graphs and in Fig. 1 and the corresponding graphic elliptic arrangements and .
These graphs appeared for the first time in [11]. They share the same Tutte polynomial, which is the following Using SAGE [1], we have computed the mixed Hodge numbers of and of and reported them in Table 1, Table 2. For this computation we have used the code available here [9];
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The server, used for computations, has been acquired thanks to the support of the University of Pisa , within the call “Bando per il cofinanziamento dell’acquisto di medio/grandi attrezzature scientifiche 2016”.
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