Colorings and flows on CW complexes, Tutte quasi-polynomials and arithmetic matroids
In this note we provide a higher-dimensional analogue of Tutte’s theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW complexes by Beck–Breuer–Godkin–Martin  and Duval–Klivans–Martin . Furthermore, we answer a question by Bajo–Burdick–Chmutov , concerning the modified Tutte–Krushkal–Renhardy polynomials defined by these authors: to this end, we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function.
The enumeration of colorings, flows and spanning trees on graphs are classical topics, unified by a two-variable polynomial due to W. T. Tutte . This polynomial specializes to both the coloring counting and the flow counting functions, and it evaluates to the number of spanning trees. H. Crapo extended Tutte’s definition to arbitrary matroids and since then this Tutte polynomial went on to become one of the most studied matroid polynomial invariants with great theoretical significance and a host of applications — e.g., in statistics and physics. Recently, this classical setup has been generalized in two ways.
First, the concept of coloring and flow has been generalized from graphs to higher dimensional objects such as simplicial complexes by Beck and Kemper  and, more generally, to CW complexes by Beck–Breuer–Godkin–Martin  and Duval–Klivans–Martin . These authors showed, among other things, that the functions counting the number of colorings and flows with values on a CW complex is a quasi-polynomial in . In a related vein, Bajo, Burdick and Chmutov  introduced a family of modified TKR polynomials that connects Kalai’s enumeration of weighted cellular spanning trees of complexes  to a class of polynomials defined by Krushkal and Renhardy  in their study of graph embeddings and to a polynomial defined by Bott .
On the other hand, in collaboration with M. D’Adderio  and with P. Brändén  the second-named author developed a theory of arithmetic matroids as “matroids decorated with a multiplicity function”, abstracting the arithmetic properties of lists of elements in finitely generated abelian groups. To each arithmetic matroid is naturally associated an arithmetic Tutte polynomial. These polynomials have been in the focus of recent and lively research, which brought to light manifold connections and a rich structure theory. For instance, arithmetic Tutte polynomials specialize to Poincaré polynomials of toric arrangements , to Ehrhart polynomials of zonotopes  and to the Hilbert series of some zonotopal spaces . Moreover, they can be recovered from the Tutte polynomials for group actions on semimatroids , and they satisfy a convolution formula .
With a list of elements in a finitely generated abelian group is also associated a Tutte quasi-polynomial , which interpolates between the (ordinary) Tutte polynomial and the arithmetic Tutte polynomial. This quasi-polynomial does not depend only on the arithmetic matroid, but on a finer structure: a matroid over in the sense of . As pointed out in , the enumerating functions of colorings and flows on a CW complex are not matroidal, and hence cannot be obtained from the ordinary Tutte polynomial. In this paper we show that, however, they are specializations of the Tutte quasi-polynomial, thus providing a higher-dimensional analogy to Tutte’s celebrated theorem for graphs .
Moreover, we show that the set of arithmetic matroids over a fixed underlying matroid has a natural structure of commutative monoid. This implies that the modified TKR polynomials are indeed arithmetic Tutte polynomials; in particular, their coefficients are positive.
In this way we address questions of the authors of [2, 3, 12], who ask whether and how the coloring and flow polynomials for CW complexes and the modified TKR polynomials are related to arithmetic matroids.
Structure of the paper
In Section 2 we start off with some preliminaries on incidence algebras and arithmetic matroids. We prove a general theorem about products of integer functions on posets (Theorem 1) and specialize it to one about products of arithmetic multiplicity functions (Theorem 2).
The first-named author has been supported by the Swiss National Foundation Professorship grant PP00P2_150552/1. We thank Yvonne Kemper for pointing out , and Fengwei Zhou for finding an error in a previous version of this paper.
2. On arithmetic matroids
2.1. Poset theory preliminaries
The goal of this section is to prove a result on Möbius functions of posets (short for “partially ordered sets”) which will serve as a stepping stone towards Theorem 2. We will assume familiarity with basic terminology of poset theory. We refer the reader unfamiliar with it to .
Throughout, we will let denote a finite poset.
The so-called Möbius function of is the function
defined recursively as follows
where for simplicity we write .
The (dual) Möbius transform
It is characterized by .
Consider two elements . If there is an element with
then is unique, called the meet (or minimal upper bound) of and , and denoted by . If every pair admits a meet, the poset is called a meet semilattice.
The following lemma should be folklore. We give here a proof for completeness, because we do not know of a reference for it.
Let be a meet-semilattice, and be a function such that for all . Then,
for all .
Define for all
We claim that
The right-to left inclusion is clear: means , hence . For the left-to-right inclusion consider . The set has a unique maximal element (since and imply – hence, imply ). Now we see that . Uniqueness of implies that the union is indeed disjoint.
Thus, for all we have
and, by Möbius inversion,
as required. ∎
Let be a meet-semilattice, and consider two functions . If and for all , then for all .
The positivity hypothesis allows us to define, for every and , a set
where the are pairwise distinct formal elements – i.e., if and only if , , . Then, set
Notice also that, by definition of ,
Consider now the family of sets defined by
Since cartesian products commute with intersections, for we have
and thus, by Lemma 1,
The claim now follows because for all .
2.2. Arithmetic matroids
In this section we recall basic definitions on matroids and arithmetic matroids in order to set some notation, and we prove Theorem 2. For background on matroid theory we refer, e.g., to Oxley’s textbook , while our presentation of arithmetic matroids follows mostly .
A matroid is given by a pair , where is a finite set and is a function such that, for all ,
A molecule in a matroid is a triple of disjoint subsets of such that, for every with ,
To the molecule , following e.g. , we associate a poset
ordered by if , .
The poset is bounded, with unique minimal element and unique maximal element . Moreover, every interval in (say, ) is the poset for another molecule (i.e., ).
Given any function and a molecule of a matroid over the ground set , we define as the function with
An arithmetic matroid is a triple where is a matroid, and is a function satisfying the following axioms.
For every molecule of
For every molecule of
For all and all ,
Axiom (P) is usually given in a different form. In fact, for a molecule we see that the poset is boolean and the length of the interval is . Therefore, the Möbius function of satisfies
If we now expand our form of axiom (P) we get
and we recover the formulation given in .
2.3. Product of multiplicity functions
Consider now a fixed matroid , two (possibly different) functions and their (pointwise) product
If both and satisfy axiom (P), so does .
Suppose and both satisfy (P) and consider a molecule . The poset is boolean, hence in particular a (meet semi-)lattice. Since every interval of defines a molecule, and satisfy the conditions of Theorem 1 on . Hence,
If both and are arithmetic matroids, then is also an arithmetic matroid.
The triple satisfies (P) by Lemma 2, and (Q), (A) trivially. ∎
This theorem endows the set of arithmetic matroids over a fixed underlying matroid with a natural product, which makes it into a commutative monoid. We leave the investigation of this algebraic structure as an open problem.
3. On Tutte quasi-polynomials associated to cell complexes
3.1. The Tutte quasi-polynomial
Let be a finitely generated abelian group, be a finite set, and be a list (multiset) of elements in . For every we denote by the sublist , by the subgroup that it generates, and by the torsion subgroup of the quotient . In [6, Section 7], the Tutte quasi-polynomial of is defined as follows.
If for every the integer is coprime with , then equals and we get the ordinary Tutte polynomial of the matroid of linear dependencies among elements of :
On the other hand, when for every the integer is a multiple of we have that is trivial and we obtain the arithmetic Tutte polynomial:
Therefore is a quasi-polynomial function that in some sense interpolates between these two polynomials. It appeared as a specialization of a multivariate ”Fortuin–Kasteleyn quasi-polynomial”.
Now recall the following definitions.
Definition 3 ([6, Section 7]).
Let , and be as above.
A proper -coloring is an element such that for all .
A nowhere zero -flow is a function such that
The number of proper -colorings and the number of nowhere zero -flows are denoted by and respectively.
The following statement generalizes a result of .
Lemma 3 ([6, Theorem 9.1]).
In particular, and are quasi-polynomial functions of , called the chromatic quasi-polynomial and the flow quasi-polynomial respectively.
3.2. On flows and colorings on CW complexes
Let be a CW complex of dimension and, for every , let be the set of the -dimensional cells of . The top-dimensional boundary map is represented by a matrix with integer entries, that (by a slight abuse of notation) we denote again by . By reducing modulo , we get a map , that we can view as a matrix with coefficients in .
Let and be as above.
a proper -coloring of is an element such that all the entries of the vector are nonzero.
a nowhere zero -flow on is an element such that the coordinate is nonzero for every .
In fact, to the (integer) matrix we can associate a Tutte quasi-polynomial, an arithmetic matroid and an arithmetic Tutte polynomial. With the following lemma we address [3, Remark 3.15] by showing that the coloring- and flow- counting quasi-polynomials of  and  are instances of the coloring- and flow- quasi-polynomials associated to the matrix .
Definitions (1’) and (2’) agree with definitions (1) and (2), when , , and .
Every uniquely extends to a homomorphism . Then since , (1) specializes to (1’). On the other hand, definition (2’) is equivalent to saying that is a function such that . This is precisely the specialization of definition (2). ∎
With the notations above, we have:
As pointed out in , the Tutte quasi-polynomial is not an invariant of the arithmetic matroid, but is an invariant of the matroid over associated to the matrix . We call this matroid the cellular matroid over of .
Underlying matroids of cellular matroids over (i.e., the matroids defined by the matrices ) have been studied in their own right. Allowing different generality for the complex one obtains different interesting classes of matroids. Already in the case where is a simplicial complex, the matroids obtained this way are strictly more general than graphical matroids .
Given a -dimensional CW-complex , for every the -skeleton of is itself a -dimensional CW-complex for which we can carry out all considerations of this section. Thus, in fact gives rise to a class of arithmetic quasi-polynomials and arithmetic matroids. In the following section we will consider properties of this class as a whole.
4. On the modified Tutte-Krushkal-Renhardy polynomial
When considering cell complexes as higher dimensional generalizations of graphs, besides flows and colorings it is natural to enumerate the analogue of spanning trees. Following Kalai , this enumeration is weighted by the cardinality of the torsion of the subcomplexes that are enumerated. This line of thought inspired , where the authors introduced a class of polynomials arising as a modification of Krushkal and Renhardy’s polynomial invariants of triangulations. This last section is devoted to answering a question of  which we will state after reviewing some definitions (following [2, 15]).
We denote by the family of all spanning subcomplexes of dimension , i.e., of all the subcomplexes such that . These are naturally identified with the subsets of , the set of -dimensional cells of the -skeleton of . Let be the -th Betti number of (i.e., the rank of the homology ), and let be the cardinality of its torsion, .
As has been pointed out e.g. in , the function is the multiplicity function of the arithmetic matroid defined by the matrix .
Definition 6 ([2, Definition 3.1]).
The -th Tutte–Krushkal–Renhardy (TKR for short) polynomial of is defined in  as
The “modified -th Tutte–Krushkal–Renhardy (TKR for short) polynomial” of is
Following [11, Section 5.4] we note that is the number of ”cellular -spanning trees” of (according to Definition 2.1 of ), while is an invariant introduced by G. Kalai in : the number of cellular -spanning trees of , each counted with multiplicity . Moreover, if and are dual cell structures on a sphere (according e.g. to [2, Definition 1.2]) we have
by , and
by [2, Theorem 3.4].
In [2, Remark 3.3] the authors ask whether the multiplicity defines an arithmetic matroid. The results established in Section 2 allow us to give a positive answer to this question.
Let be a CW-complex of dimension and, for every let denote the cellular matroid of the -skeleton of (see Remark 6). Then, for every the pair (, ) is an arithmetic matroid, and the modified -th Tutte–Krushkal–Renhardy polynomial is the associated arithmetic Tutte polynomial. In particular, the coefficients of are nonnnegative.
- This will avoid unnecessary technicalities and will suffice for the applications later in the paper, even though most of what we will prove in this section holds in the generality of locally finite posets.
- We will henceforth simply use the term Möbius transform. It is referred to as “dual form” in [20, Proposition 3.7.2]
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