Poincaré polynomial of elliptic arrangements is not determined by the Tutte polynomial

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Abstract

The Poincaré polynomial of the complement of an arrangements in a non compact group G is a specialization of the G-Tutte polynomial associated with the arrangement. In this article we show two unimodular elliptic arrangements (built up from two graphs) with the same Tutte polynomial, having different Betti numbers.

Introduction

Let AM(k,n;Z) be an integer matrix and let G be a group of the form H×(S1)p×Rq, where H is a finite abelian group. Each column α of A defines a morphism from Gk to G given by (g1,,gk)α1g1+α2g2++αkgk.We call HiGk the kernel of the map defined by the ith-column of A. The complement of the arrangement A in G is the topological space M(A;G)=Gki=1nHi.

When G=R2 we obtain the definition of central hyperplane arrangements (with rational equations). If G=S1×R the arrangement is called toric. We are mainly interested in the case G=S1×S1E (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 G-Tutte polynomial [7], [8], [12]. Given a subset S of [n]={1,2,,n} we call layer any connected component of the intersection iSHi. The poset of layers is the set of all layers ordered by reverse inclusion. The arithmetic matroid is the triple ([n],rk,mG) associated with toric, hyperplanes or elliptic arrangements, where rk(S) and mG(S) are, respectively, the codimension and the number of connected components of iSHi. The G-Tutte polynomial is a generalization of the arithmetic Tutte polynomial and of the classical Tutte polynomial; it is defined by TAG(x,y)=defS[n]mG(S)(x1)rk[n]rk(S)(y1)|S|rk(S)where mG is the multiplicity function defined in [7, Definition 4.6]; if G is connected, then mG(S) coincides with the number of connected components of iSHi.

Recently, a formula for the Poincaré polynomial of M(A;G) was found by Liu, Tran and Yoshinaga [7] when G is not compact, i.e. q>0. This formula involves the G-characteristic polynomial χAG(t), which is a specialization of the G-Tutte polynomial: χAG(t)=(1)rk[n]tkrk[n]TAG(1t,0).When G is not compact, the Poincaré polynomial of M(A;G) is PM(A;G)=(tp+q1)kχAGPG(t)tp+q1,where PG(t)=mG()(t+1)p is the Poincaré polynomial of the group G. The formula e(M(A;G))=(1)(p+q)kχAG((1)p+qe(G))for the Euler characteristic holds for all groups G (e(G) is the Euler characteristic of G), see [2], [7].

We focus on the “smallest” compact group G=S1×S1, the case G=S1 being trivial. From now on, we denote the two-dimensional compact torus S1×S1 by E. 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 M(A;E)Ek. As shown in [10], this model is combinatorial, i.e. can be defined from the arithmetic matroid ([n],rk,mE). 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 Ek+1EEk be the quotient of Ek+1 by the diagonal action of E. Given a finite graph G=([k+1],E), undirected and without loops or multiple edges, we can define an arrangement AG in Ek+1E given by the divisors He=Hi,j=def{g̲Ek+1Egi=gj},for each edge e=(i,j)E. We fix arbitrarily a spanning forest T of G and an orientation of G. Consider the external

The example

Consider the two graphs G1 and G2 in Fig. 1 and the corresponding graphic elliptic arrangements A1 and A2.

These graphs appeared for the first time in [11]. They share the same Tutte polynomial, which is the following T(x,y)=x7+4x6+x5y+9x5+6x4y+3x3y2+x2y3+13x4+13x3y+7x2y2+3xy3+y4+12x3+15x2y+9xy2+3y3+7x2+9xy+4y2+2x+2y.Using SAGE [1], we have computed the mixed Hodge numbers of M(A1) and of M(A2) 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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