Betti numbers are testable

Extremal Betti numbers of Rips complexes

Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti numbers are given.

Limit theorems for Betti numbers of random simplicial complexes

There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems known so far about limiting distributions of random Betti numbers. In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complex: Erdős-Rényi random clique complexes, random Vietoris-Rips complexes, and random Čech complexes. These results may be of practical interest in to…

Betti numbers of unordered configuration spaces of small graphs

The purpose of this document is to provide data about known Betti numbers of unordered configuration spaces of small graphs in order to guide research and avoid duplicated effort. It contains information for connected multigraphs having at most nine edges which contain no loops, no bivalent vertices, and no internal (i.e., non-leaf) bridges.

Extremal problems related to Betti numbers of flag complexes

We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32^n for the sum of Betti numbers of any n-vertex flag complex and 1.25^n for the independence complex of a triangle-free graph. These findings imply upper bounds for the Betti numbers of various related classes of spaces, including the neighbourhood complex of a graph. We also make some related observations.

Multigraded Betti numbers of some path ideals

We determine (multi)graded Betti numbers of path ideals of lines and star graphs.

Toric h-vectors and Chow Betti Numbers of Dual Hypersimplices

The toric h-numbers of a dual hypersimplex and the Chow Betti numbers of the normal fan of a hypersimplex are the ranks of intersection cohomology and Chow cohomology respectively of the torus orbit closure of a generic point in the Grassmannian. We give explicit formulas for these numbers. We also show that similar formulas hold for the coordinator numbers of type A^* lattices.

On the extremal Betti numbers of binomial edge ideals of block graphs

We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.

小Betti数的三次环面障碍

Cubic torus obstructions of small Betti number

本文针对图在环面上的可嵌入性问题，给出了Betti数不超过八的三次环面障碍的完整理论分类。

Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids

To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid.

Diffeomorphic moment-angle manifolds with different Betti numbers

We describe here two simple polytopes that have different Betti numbers and whose moment-angle manifolds are diffeomorphic.

On the extremal Betti numbers of the binomial edge ideal of closed graphs

We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${\rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${\rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal

Extremal Betti numbers of some classes of binomial edge ideals

Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.

多面体的空单形与分次贝蒂数

Empty simplices of polytopes and graded Betti numbers

Kalai，Kleinschlaud和Lee关于单纯多胞形的空单形个数的猜想是通过将它与多胞形的第一分次贝蒂数联系起来而建立的。证明使我们能够获得明确的最佳界限的数量空单形的任何给定的尺寸。作为一个关键结果，我们证明了标准分次K-代数的分次贝蒂数关于其Hilbert函数的最优界。

Extremal Betti numbers of edge ideals

Given integers $r$ and $b$ with $1 \leq b \leq r$, a finite simple connected graph $G$ for which ${\rm reg}(S/I(G)) = r$ and the number of extremal Betti numbers of $S/I(G)$ is equal to $b$ will be constructed.

Extremal Betti numbers of graded modules

Let $S$ be a polynomial ring in $n$ variables over a field $K$ of characteristic $0$. A numerical characterization of all possible extremal Betti numbers of any graded submodule of a finitely generated graded free $S$-module is given.

The Betti numbers of regular Hessenberg varieties are palindromic

Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs describing a representation of the symmetric group on the cohomology of regular semisimple Hessenberg varieties for $GL_n(\mathbb{C})$. A key component of their argument is that the Betti numbers of regular Hessenberg varieties for $GL_n(\mathbb{C})$ are palindromic. In this paper, we extend this result to all reductive algebraic groups, proving that the Betti numbers of regular Hessenberg varieties are palindromic.

Betti numbers of weighted oriented graphs

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we charac…

Dependence of Betti Numbers on Characteristic

We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with componentwise linear resolutions. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals. We also prove that the Betti table of a monomial ideal over the field of rational numbers can be obtained from the Betti table over any field by a sequence o…

Graded Betti numbers of cycle graphs and standard Young tableaux

We give a bijective proof that the Betti numbers of a minimal free resolution of the Stanley-Reisner ring of a cycle graph (viewed as a one-dimensional simplicial complex) are given by the number of standard Young tableaux of a given shape.

Non-vanishingness of Betti numbers of edge ideals

Given finite simple graph one can associate the edge ideal. In this paper we discuss the non-vanishingness of the graded Betti numbers of edge ideals in terms of the original graph. In particular, we give a necessary and sufficient condition for a chordal graph on which the graded Betti number does not vanish and characterize the graded Betti number for a forest. Moreover we characterize the projective dimension for a chordal graph.

对称迭代贝蒂数

Symmetric iterated Betti numbers

定义了多项式环中齐次理想$I$的一组不变量，称为$I$的对称迭代贝蒂数.对于$I_Γ$，单纯复形$Γ$的Stanley-Reisner理想，这些数是Duval和Rose引入的$Γ$的外迭代贝蒂数的对称对应。我们发现，对称迭代贝蒂数的理想$I$符合一个特定的反向字典通用初始理想$\Gin(I)$的$I$，并解释这些不变量的相关素数和标准对的$\Gin(I)$。证明了对于理想的$I=I_Γ$，$I_Γ$的极值贝蒂数恰好是$Γ$的极值（对称或外）迭代贝蒂数.最后给出了关于单纯复形的对称数与外迭代贝蒂数之间关系的一些结果和图解。

超图的贝蒂数

Betti numbers of hypergraphs

本文研究超图的一些代数性质，特别是超图的贝蒂数.我们定义了一些不同类型的完全超图，据我们所知，以前没有考虑在文献中。同时，我们以一种自然的方式在超图上定义了一个积，它在某种意义上与单纯复形上的并运算是对偶的。对于这类乘积，我们给出了贝蒂数的一般公式，它在线性分解的情况下有很好的特殊性。

森林的贝蒂数

The Betti numbers of forests

本文给出了一类Stanley-Reisner理想（与森林有关的图理想）的贝蒂数的递推公式.这给出了这些理想的投射维数的一个纯组合的定义，它是森林的一个新的数值不变量。最后，我们提出了一个可能的扩展，这个不变量一般图。

图理想的贝蒂数的特征独立性

Characteristic-independence of Betti numbers of graph ideals

本文研究了2次生成的Stanley-Reisner理想的贝蒂数.我们证明了前六个贝蒂数不依赖于基域的特性。我们还表明，如果变量$n$的数量是最多10，所有的贝蒂数是独立的基域。对于$n=11$，精确地存在4个例子，其中贝蒂数取决于基域。这等价于至多10个顶点的旗复形的同调是无挠的，且恰好存在4个11个顶点的非同构旗复形的同调有挠。在上面提到的4个例子中，第8个贝蒂数依赖于基域，因此我们得出结论，总是独立于基域的最高贝蒂数是6或7;如果前者为真，那么我们证明了一定存在一个12个顶点的图，其第7个贝蒂数依赖于基域。

Graded Expectations: Betti numbers and anti-lecture hall compositions of random threshold graphs

This paper examines the one-to-one-to-one correspondence between threshold graphs, Betti numbers of quotients of polynomial rings by $2$-linear ideals, and anti-lecture hall compositions. In particular, we establish new explicit combinatorial mappings between each of these classes of objects and calculate the expected values of the Betti numbers and anti-lecture hall composition corresponding to a random threshold graph.

非常面映射的多格Betti数

Multigraded Betti numbers of Veronese embeddings

本文研究非常面嵌入的多格Betti数，利用Hochster公式和离散Morse理论推导其消失与非消失结果。

Bounds for the Betti numbers of successive stellar subdivisions of a simplex

We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. The bound depends only on the number of subdivisions, and we construct examples which prove that it is sharp.

么半群复形的Gröbner基与贝蒂数

Gröbner bases and Betti numbers of monoidal complexes

本文考虑幺半群复形及其相应的代数，称为复环面环。这些环推广了Stanley-Reisner环和仿射幺半群代数。计算复曲面环的表现理想的初始理想，并确定其分次贝蒂数.我们的结果推广了组合交换代数中著名的Hochster定理.

单项理想与移位斜形的贝蒂数

Betti numbers of monomial ideals and shifted skew shapes

本文提出了两个新的关于定度生成的单项理想的分解贝蒂数的下界问题。第一个问题涉及任何这样的理想和边界的总贝蒂数，而第二个问题的理想是二次和双齐次相对于两个变量集，但给出了一个更精细的分级下界。这些问题解决了某些类的理想，推广（在两个不同的方向）的边缘理想的阈值图和费勒斯图。在这个过程中，我们产生特别简单的细胞的线性决议强稳定和squarefree强稳定的理想产生在一个固定的程度，和组合的解释贝蒂数的其他类的理想，所有这些都是独立的系数字段。

Betti numbers associated to the facet ideal of a matroid

To a matroid M with n edges, we associate the so-called facet ideal F(M) generated by monomials corresponding to bases of M. We show that the Betti numbers related to an N-graded minimal free resolution of F(M) are determined by the Betti numbers related to the blocks of M. Similarly, we show that the higher weight hierarchy of M is determined by the weight hierarchies of the blocks, as well. Drawing on these results, we show that when M is the cycle matroid of a cactus graph, the Betti numbers determine the higher weight hierarchy -- and vice versa. Finally, we demonstrate by way of counterex…