Poincaré polynomial of elliptic arrangements is not a specialization of the Tutte polynomial
aa r X i v : . [ m a t h . A T ] O c t POINCAR´E POLYNOMIAL OF ELLIPTICARRANGEMENTS IS NOT A SPECIALIZATION OF THETUTTE POLYNOMIAL
ROBERTO PAGARIA
Abstract.
The Poincar´e polynomial of the complement of an arrange-ments in a non compact group is a specialization of the G -Tutte poly-nomial associated with the arrangement. In this article we show twounimodular elliptic arrangements (built up from two graphs) with thesame Tutte polynomial, having different Betti numbers. Introduction
Let A ∈ M( k, n ; Z ) be an integer matrix and let G be a group of the form H × ( S ) p × R q , where H is a finite abelian group. Each column α of A defines a morphism from G k to G given by( g , . . . , g k ) α g + α g + · · · + α k g k . We call H i ⊂ G k the kernel of the map defined by the i th -column of A . The complement of the arrangement A in G is the topological space M ( A ; G ) = G k \ n [ i =1 H i . When G = R ≃ C we obtain the classical definition of hyperplane ar-rangements . If G = S × R ≃ C ∗ the arrangement is called toric . We aremainly interested in the case G = S × S ≃ E (an elliptic curve), thisarrangements is called elliptic arrangement .There are several combinatorial objects associated with an arrangement:for instance, the poset of layers, the arithmetic matroid ([2, 3]) and the G -Tutte polynomial ([7, 6, 8]). Given a subset S of [ n ] = { , , . . . , n } we call layer any connected component of the intersection T i ∈ S H i . The poset oflayers is the set of all layers ordered by reverse inclusion. The arithmeticmatroid is the triple ([ n ] , rk , m G ) associated with toric, hyperplanes or ellip-tic arrangements, where rk( S ) and m G ( S ) are, respectively, the codimensionand the number of connected components of T i ∈ S H i . The G -Tutte poly-nomial is a generalization of the arithmetic Tutte polynomial and of theclassical Tutte polynomial; it is defined by T GA ( x, y ) def = X S ⊆ [ n ] m G ( S )( x − rk[ n ] − rk( S ) ( y − | S |− rk( S ) . Recently, a formula for the Poincar´e polynomial of M ( A ; G ) was found byLiu, Tran and Yoshinaga [6] when G is not compact, i.e. q >
0. This formula involves the G -characteristic polynomial χ GA ( t ), which is a specialization ofthe G -Tutte polynomial: χ GA ( t ) = ( − rk[ n ] t k − rk[ n ] T GA (1 − t, . When G is not compact, the Poincar´e polynomial of M ( A ; G ) is P M ( A ; G ) = ( − t p + q − ) k χ GA (cid:18) − P G ( t ) t p + q − (cid:19) , where P G ( t ) = m G ( ∅ )( t + 1) p is the Poincar´e polynomial of the group G .The formula e ( M ( A ; G )) = ( − ( p + q ) k χ GA (( − p + q e ( G ))for the Euler characteristic holds for all groups G ( e ( G ) is the euler charac-teristic of G ), see [1, 6].We focus on the “smallest” compact group G = S × S , the case G = S being trivial. From now on, we denote the two-dimensional compact torus S × S by E . In this case, Bibby [1] and Dupont [5] have given a modelof the cohomology ring H • ( M ( A ; E ); Q ), provided by the second page ofthe Serre spectral sequence for the inclusion M ( A ; E ) ֒ → E k . As shownin [9], this model is combinatorial, i.e. can be defined from the arithmeticmatroid ([ n ] , rk , m E ). Thus the Betti numbers are implicitly codified inthe arithmetic matroid, but there is no explicit formula that allows theircalculation. We will show that these Betti number are independent fromthe arithmetic Tutte polynomial, exhibiting an example.2. The model for cohomology
We recall the model developed by Dupont [5] and Bibby [1] for the coho-mology ring in the particular case of graphic elliptic arrangements.Let E k +1 /E ≃ E k be the quotient of E k +1 by the diagonal action of E . Given a finite graph G = ([ k + 1] , E ), undirected and without loops ormultiple edges, we can define an arrangement A G in E k +1 /E given by thedivisors H e = H i,j def = { g ∈ E k +1 /E | g i = g j } , for each edge e = ( i, j ) ∈ E . We fix arbitrarily a spanning forest T of G andan orientation of G .Consider the external algebra Λ over the rationals on the generators { ω e , x t , y t } e ∈E t ∈T . We set the bi-degree of ω e to be (1 ,
0) and the one of x t and y t to be (0 , e = i → j the element x e = P f ∈ γ ( e ) ǫ ( e, f ) x f , where γ ( e ) isthe unique path from i to j in T and ǫ ( e, f ) is 1 if the arc f is orientedas in the path γ ( e ), − I ⊂ Λgenerated by the elements ω e x e , ω e y e and l X i =0 ( − i ω e ω e . . . b ω e i . . . ω e l , for every cycle C = ( e , e , . . . , e l ) of G . We call E ( A G ) the quotient Λ /I .Finally, we define the differential d : E ( A G ) → E ( A G ) on the generatorsby d ( ω e ) = x e y e and d ( x e ) = d ( y e ) = 0. This is well defined since OINCAR´E POLYNOMIAL OF ELLIPTIC ARRANGEMENTS 3
Figure 1.
The graph G on the left and G on the rightd ( I ) ⊆ I . The model ( E ( A G ) , d ) coincides with the second page of theLeray spectral sequence and the cohomology of the second page (i.e. the thirdone) is the cohomology ring of M ( A G ; E ) with rational coefficients. Thebi-gradation of the third page corresponds to the bi-gradation given by themixed Hodge structure (and the total degree). Let e ( a, b ) be the dimension ofthe homogeneous subspace of bi-degree ( a, b ) of the third page. The number e ( a, b ) coincides with the dimension of the subspace of H a + b ( M ( A G ; E )) ofweight a + 2 b (see [4, 4 pg 81]).Since the elliptic arrangement A G is unimodular , i.e. every subset of divi-sors has connected intersection, the G -Tutte polynomial T EA G coincides withthe classical Tutte polynomial T G associated with the graph G . In particularthe dimension of E a,b ( A G )can be easily calculated from X a,b dim E a,b ( A G ) t a s b = T G (cid:18) t ) s , (cid:19) s k . Thus, the Hodge polynomial evaluated in ( − , u ) is X n X m ≥ ( − m e ( n − m, m ) u n = T G (cid:18) − (1 + u ) u , (cid:19) ( − u ) k , and the Euler characteristic of M ( A G ; E ) is ( − k T G (1 , The example
Consider the two graphs G and G in fig. 1 and the corresponding graphicelliptic arrangements A and A . These graphs appeared for the first timein [10]. They share the same Tutte polynomial, which is the following T ( x, y ) = x + 4 x + x y + 9 x + 6 x y + 3 x y + x y + 13 x + 13 x y ++7 x y + 3 xy + y + 12 x + 15 x y + 9 xy ++3 y + 7 x + 9 xy + 4 y + 2 x + 2 y. Using SAGE [11], we have computed the mixed Hodge numbers of M ( A )and of M ( A ) and reported them in Tables 1 and 2. For this computationwe have used the code available here ; the calculation of the Hodge number e (4 ,
2) has taken more than 2 days with a CPU of 2 . http://poisson.phc.dm.unipi.it/~pagaria/Graphic_Elliptic_Arr.txt R. PAGARIA
Table 1.
The Hodge numbers of M ( A )0 40 6 260 4 45 740 8 54 154 1160 6 60 200 259 940 2 29 144 302 224 411 14 80 234 358 260 77 8In position ( i, j ) there is the dimension of the subgroup ofweight i + 2 j in H i + j ( M ( A ; E )). Table 2.
The Hodge numbers of M ( A )0 40 6 260 6 45 740 10 69 162 1160 6 59 202 271 1000 2 30 150 301 224 381 14 80 234 359 266 77 8In position ( i, j ) there is the dimension of the subgroup ofweight i + 2 j in H i + j ( M ( A ; E )).of RAM. Some Hodge numbers have being calculated using the followingformula X n X m ≥ ( − m e ( n − m, m ) u n = 1 + 14 u + 80 u + 232 u + 329 u ++ 122 u − u − u + 164 u − u − u + 68 u − u + 4 u . The Poincar´e polynomials of M ( A ) and M ( A ) are different: P M ( A ; E ) ( t ) = 1 + 14 t + 82 t + 269 t + 570 t + 820 t + 765 t + 363 t ,P M ( A ; E ) ( t ) = 1 + 14 t + 82 t + 270 t + 578 t + 844 t + 785 t + 366 t . The Euler characteristic of M ( A ; E ) and of M ( A ; E ) are both equal to − Acknowledgements.
The server, used for computations, has been ac-quired thanks to the support of the University of Pisa, within the call“Bando per il cofinanziamento dell’acquisto di medio/grandi attrezzaturescientifiche 2016”.
References [1] Christin Bibby. Cohomology of abelian arrangements.
Proc. Amer. Math. Soc. ,144(7):3093–3104, 2016.
OINCAR´E POLYNOMIAL OF ELLIPTIC ARRANGEMENTS 5 [2] Petter Br¨and´en and Luca Moci. The multivariate arithmetic Tutte polynomial.
Trans.Amer. Math. Soc. , 366(10):5523–5540, 2014.[3] Michele D’Adderio and Luca Moci. Arithmetic matroids, the Tutte polynomial andtoric arrangements.
Adv. Math. , 232:335–367, 2013.[4] Pierre Deligne. Poids dans la cohomologie des vari´et´es alg´ebriques.
Proceedings of theInternational Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1 , pages79–85, 1975.[5] Cl´ement Dupont. The Orlik-Solomon model for hypersurface arrangements.
Ann.Inst. Fourier (Grenoble) , 65(6):2507–2545, 2015.[6] Ye Liu, Tan Nhat Tran, and Masahiko Yoshinaga. G -Tutte polynomials and abelianLie group arrangements. ArXiv e-prints , page 32, July 2017.[7] Luca Moci. A Tutte polynomial for toric arrangements.
Trans. Amer. Math. Soc. ,364(2):1067–1088, 2012.[8] Tan Nhat Tran and Masahiko Yoshinaga. Combinatorics of certain abelian Lie grouparrangements and chromatic quasi-polynomials.
ArXiv e-prints , page 16, May 2018.[9] Roberto Pagaria. Configuration spaces of points in an elliptic curve.
ArXiv e-prints ,page 35, May 2018.[10] Werner Schw¨arzler. Being Hamiltonian is not a Tutte invariant.
Discrete Math. ,91(1):87–89, 1991.[11] The Sage Developers.
SageMath, the Sage Mathematics Software System (Version8.2) , 2018. . Roberto Pagaria
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italia
E-mail address ::