Edna F Jasso-Hernandez, Yongwu Rong
For each graph, we construct a bigraded chain complex whose graded Euler characteristic is a version of the Tutte polynomial. This work is motivated by earlier work of Khovanov, Helme-Guizon and Rong, and others.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
For each graph and each positive integer $n$, we define a chain complex whose graded Euler characteristic is equal to an appropriate $n$-specialization of the dichromatic polynomial. This also gives a categorification of $n$-specializations of the Tutte polynomial of graphs. Also, for each graph and integer $n\le 2$, we define the different one variable $n$-specializations of the dichromatic polynomials, and for each polynomial we define graded chain complex whose graded Euler characteristic is equal to that polynomial. Furthermore, we explicitly categorify the specialization of the Tutte poly…
Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobas-Riordan polynomial (a ge…
This is a survey recent works on topological extensions of the Tutte polynomial.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We generalize the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollobás-Riordan-Tutte polynomial.
We introduce and study the notion of the $G$-Tutte polynomial for a list $\mathcal{A}$ of elements in a finitely generated abelian group $Γ$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite abelian groups to $G$. The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy…
It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Ou…
A graph $G$ is said to be $p$-periodic, if the automorphism group $Aut(G)$ contains an element of order $p$ which preserves no edges. In this paper, we investigate the behavior of graph polynomials (Negmai and Tutte) with respect to graph periodicity. In particular, we prove that if $p$ is a prime, then the coefficients of the Tutte polynomial of such a graph satisfy a certain necessary condition. This result is illustrated by an example where the Tutte polynomial is used to rule out the periodicity of the Frucht graph.
Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality…
Weighted Tutte-Grothendieck polynomials of graphs
本文引入图的加权（调和）色多项式与Tutte-Grothendieck多项式概念，推广了配方定理，构造了相关不变量，并探讨了范畴化问题。
A categorification for the partial-dual genus polynomial
本文利用与无定向拓扑量子场论相关的扩展Frobenius代数，为带状图的部分对偶亏格多项式提供了一种范畴化构造。
The Bollobás-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly-colored stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs, which obeys a contraction/deletion recursion relation similar to that of the Tutte and Bollobás-Riordan polynomials. However, it is defined on a much broader …
For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. This work is motivated by Khovanov's work on categorification of the Jones polynomial of knots.
Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summation for the bracket. The categorifications are based on new gradings associated with these arrow numbers, and give homology theories associated with oriented virtual knots and links via extra structure on the Khovanov chain complex. Applications are given to the estimation of vi…
In this paper, we characterize the sigma-adequacy of a link diagram in two ways: in terms of a certain edge subset of its Tait graph and in terms of a certain product of Tutte polynomials. Furthermore, we show that the symmetrized Tutte polynomial of the Tait graph of a link diagram can be written as a sum of these products of Tutte polynomials, where the sum is over the sigma-adequate states of the given link diagram. Using this state sum, we show that the number of sigma-adequate states of a link diagram is bounded above by the number of spanning trees in its associated Tait graph. By combin…
We transform Tutte-Grothedieck invariants thus also Tutte polynomials on matroids so that the contraction-deletion rule for loops (isthmuses) coincides with the general case.
In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.
Categorification of some Penrose polynomials
通过构造带边曲面的TQFT值立方分解，得到带状图的双重和三重分次Penrose型同调，其欧拉特征恢复某些Penrose多项式的特例，并细化经典Penrose准则。
Extreme coefficients of multiplicity Tutte polynomials
本文推导了重数Tutte多项式的一般系数和六个极端系数的表达式，并作为特例得到了经典Tutte多项式的相应系数表达式。
We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective.
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.
By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the literature arise from this framework. These polynomials include the classical Tutte polynomial of graphs and matroids, Las Vergnas' Tutte polynomial of the morphism of matroids and his Tutte polynomial for embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal polynomial, and the Penrose polynomial. We show that our Tutte polynomials of…
We give a generating set for linear relations on Tutte polynomials of rank-$r$ size-$n$ freedom matroids.
We study the computation of the Tutte polynomials of fan-like graphs and obtain expressions of their Tutte polynomials via generating functions. As applications, Tutte polynomials, in particular, the number of spanning trees, of two kinds of benzenoid systems, i.e. pyrene chains and triphenylene chains, are obtained.
Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of (V,E) and its transpose (E,V) agree. When G is a complete bipartite graph, our result recovers a well known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read of…
We prove that the Ehrhart polynomial of a zonotope is a specialization of the multiplicity Tutte polynomial. We derive some formulae for the volume and the number of integer points of the zonotope.
We give a convolution formula for the Tutte polynomial T_M (x,y) of a matroid M.
We follow the example of Tutte in his construction of the dichromate of a graph (that is, the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, Bollobás-Riordan polynomial and Kruskhal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking v…
This article computes the Tutte polynomial of the hyperplane arrangements associated to the complex reflection groups. The calculations are based on both formulas of De Concini and Procesi for Tutte polynomial and the normaliser of parabolic subgroups in complex reflection groups determined by Krishnasamy and Taylor.
A celebrated result of F. Jaeger states that the Tutte polynomial of a planar graph is determined by the HOMFLY polynomial of an associated link. Here we are interested in the converse of this result. We consider the question `to what extent does the Tutte polynomial determine the HOMFLY polynomial of any knot?' We show that the HOMFLY polynomial of a knot is determined by Tutte polynomials of plane graphs associated to the knot.
We establish a relation between the Bollobas-Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.
The Tutte polynomial is a well-studied invariant of graphs and matroids. We first extend the Tutte polynomial from graphs to hypergraphs, and more generally from matroids to polymatroids, as a two-variable polynomial. Our definition is related to previous works of Cameron and Fink and of Kálmán and Postnikov. We then define the universal Tutte polynomial $\T_n$, which is a polynomial of degree $n$ in $2+(2^n-1)$ variables that specializes to the Tutte polynomials of all polymatroids (hence all matroids) on a ground set with $n$ elements. The universal polynomial $\T_n$ admits three kinds of sy…
The Tutte polynomial determines the linear coefficient of the Ehrhart polynomial
本研究证明了Tutte多项式与Ehrhart多项式线性系数的关系公式，刻画了对应极值拟阵并证明了Ferroni的猜想。
The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This article has two objectives: On one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we intro…
Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems of types A,B,C, and D, with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a finite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.
A coarse Tutte polynomial for hypermaps
本文提出了超地图的Tutte多项式，通过删除-收缩递归定义，统一推广了图与嵌入图的Tutte多项式，并建立了对偶恒等式及与其他超地图多项式的关系。
We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte.
In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized,…
Myopic Tutte polynomials and Khovanov homology in $\mathbb{R}P^3$
本文提出近视 Tutte 多项式，恢复广义 Krushkal 多项式及 Jones 多项式类比，证明 Kauffman-Murasugi-Thistlethwaite 定理类比，并构建 Khovanov 同调的生成树模型，证明其由 Jones 多项式和符号决定。
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.
Tutte polynomials of matroids as universal valuative invariants
本文完整分类了使Tutte多项式成为泛赋值不变量的对偶与子式封闭拟阵族，并由此导出拟阵Tutte多项式的新关系，证明其可唯一表示为图Schubert拟阵Tutte多项式的整系数组合。
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal structure of a graph, and contains even other specializations from other sciences like the Jones polynomial in Knot theory, the partition function of the Pott model in statistical physics, and the reliability polynomial in network theory. In this article, we study the Tutte polynomial associated to more general objects which are the arrangements of hyperplanes.…
We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then reduced to certain related enumerative questions. As a consequence, we obtain new formulas for the generating functions enumerating alternating trees, labelled trees, semiorders and Dyck paths.
Multivariate Tutte polynomials of semimatroids
该研究引入半拟阵的多元Tutte等多种多项式，建立其删除-收缩递推、基活动展开及卷积恒等式，将图与拟阵Tutte多项式的相关经典结果推广到半拟阵领域。
The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial is equivalent to the Tutte polynomial. We explore various properties, expansions, specializations, and generalizations of the B-polynomial, and try to answer the following questions: 1. what properties o…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a co…
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful finite field method for this compu…
Brylawski proved the coefficients of the Tutte polynomial of a matroid satisfy a set of linear relations. We extend these relations to a generalization of the Tutte polynomial that includes greedoids and antimatroids. This leads to families of new identities for antimatroids, including trees, posets, chordal graphs and finite point sets in $\mathbb{R}^n$. It also gives a "new" linear relation for matroids that is implied by Brylawski's identities.
This paper is based on a series of talks given at the Patejdlovka Enumeration Workshop held in the Czech Republic in November 2007. The topics covered are as follows. The graph polynomial, Tutte-Grothendieck invariants, an overview of relevant elementary finite Fourier analysis, the Tutte polynomial of a graph as a Hamming weight enumerator of its set of tensions (or flows), and a description of a family of polynomials containing the graph polynomial which yield Tutte-Grothendieck invariants in a similar way.
Deletion-Contraction and the Surface Tutte Polynomial
本文统一源于曲面 Tutte 多项式与 Hopf 代数典范 Tutte 多项式的两族拓扑 Tutte 多项式，从而给出曲面 Tutte 多项式的删除-收缩定义及局部流、张力计数递归。
The Tutte polynomial is a well-studied invariant of matroids. The polymatroid Tutte polynomial $\mathcal{J}_{P}(x,y)$, introduced by Bernardi et al., is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this paper, we first prove that $\mathcal{J}_{P}(x,t)$ and $\mathcal{J}_{P}(t,y)$ are interpolating for any fixed real number $t\geq 1$, and then we study the coefficients of high-order terms in $\mathcal{J}_{P}(x,1)$ and $\mathcal{J}_{P}(1,y)$. These results generalize results on interior and exterior polynomials of hypergraphs.
We define and study "semimatroids", a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. We show that geometric semilattices are precisely the posets of flats of semimatroids. We define and investigate the Tutte polynomial of a semimatroid. We prove that it is the universal Tutte-Grothendieck invariant for semimatroids, and we give a combinatorial interpretation for its non-negative coefficients.
This is a chapter destined for the book "Handbook of the Tutte Polynomial". The chapter is a composite. The first part is a brief introduction to Orlik-Solomon algebras. The second part sketches the theory of evaluative functions on matroid base polytopes and in particular, the G-invariant (as the subject is known late 2015). A third very short section is on Hopf-algebra or coalgebra structures in Tutte polynomial theory.
We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specializations of this identity. Combinatorial or probabilistic interpretations are given for the specialized identities.
Let G be a graph with adjacency matrix A(G). Consider the matrix IA(G)=(I | A(G)), where I is the identity matrix, and let M(IA(G)) be the binary matroid represented by IA(G). Then suitably parametrized versions of the Tutte polynomial of M(IA(G)) yield the interlace polynomials of G, introduced by Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584]. Interlace polynomials subsequently introduced by other authors may be obtained from parametrized Tutte polynomials of the binary matroid represented by (I | A(G) | I+A(G)).
Convolution formulas for multivariate arithmetic Tutte polynomials
本研究针对算术拟阵的多元算术Tutte多项式，给出两个算术拟阵乘积及单个算术拟阵对应的卷积公式，可纯组合证明已有相关结论且证明更短，还得到算术拟阵特征多项式的卷积公式。
In the literature can be found several descriptions of the Tutte polynomial of graphs. Tutte defined it thanks to a notion of activity based on an ordering of the edges. Thereafter, Bernardi gave a non-equivalent notion of the activity where the graph is embedded in a surface. In this paper, we see that other notions of activity can thus be imagined and they can all be embodied in a same notion, the $Δ$-activity. We develop a short theory which sheds light on the connections between the different expressions of the Tutte polynomial.
We construct a new polynomial invariant of maps (graphs embedded in a compact surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollobás--Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.
We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of alternating sign with a combinatorial interpretation closely tied to the Dawson partition. Our definition extends in a straightforward way to polymatroids, and in this setting our polynomial has Kálmán's internal and external activity polynomials as its univariate specialisations.
We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial.
The tensor product $(G_1,G_2)$ of a graph $G_1$ and a pointed graph $G_2$ (containing one distinguished edge) is obtained by identifying each edge of $G_1$ with the distinguished edge of a separate copy of $G_2$, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored graphs and the generalized Tutte polynomial introduced by Bollobás and Riordan. In this paper we generalize the colored tensor product formula to relative Tutte polynomials of relative graphs…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。