arXiv math/06091402006-09-05math.AT

Homology of planar polygon spaces

Michael Farber, Dirk Schuetz

英文摘要：

In this paper we study topology of the variety of closed planar polygons with given side lengths. We describe the Betti numbers of the moduli spaces as functions of the length vector. We also find sharp upper bounds on the sum of Betti numbers of the moduli space depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

On the cohomology classes of planar polygon spaces

We obtain an explicit formula for the Poincare duality isomorphism H^{n-3}(Mbar(ell)) to Z/2 for the space of isometry classes of n-gons with specified side lengths, if ell is monogenic in the sense of Hausmann-Rodriguez. This has potential application to topological complexity.

2016-04-14math.AT

Manifold properties of planar polygon spaces

We prove that the tangent bundle of a generic space of planar n-gons with specified side lengths, identified under isometry, plus a trivial line bundle is isomorphic to (n-2) times a canonical line bundle. We then discuss consequences for orientability, cobordism class, immersions, and parallelizability.

2018-05-08math.AT

Topological complexity of planar polygon spaces with small genetic code

We determine lower bounds for the topological complexity of many planar polygon spaces mod isometry. With very few exceptions, the upper and lower bounds given by dimension and cohomology considerations differ by 1. This is true for 130 of the 134 generic 7-gon spaces. Our results apply to spaces of n-gons for all n, but primarily for those whose genetic codes, in the sense of Hausmann and Rodriguez, are moderately small.

2016-01-20math.AT

The Borsuk-Ulam theorem for planar polygon spaces

The moduli space of planar polygons with generic side lengths is a closed, smooth manifold. Mapping a polygon to its reflected image across the $X$-axis defines a fixed-point-free involution on these moduli spaces, making them into free $\mathbb{Z}_2$-spaces. There are some important numerical parameters associated with free $\mathbb{Z}_2$-spaces, like index and coindex. In this paper, we compute these parameters for some moduli spaces of polygons. We also determine for which of these spaces a generalized version of the Borsuk-Ulam theorem hold. Moreover, we obtain a formula for the Stiefel-Wh…

2022-07-22math.AT

Topological complexity of some planar polygon spaces

Using known results about their mod-2 cohomology ring, we prove that the topological complexity of the space of isometry classes of n-gons in the plane with one side of length r and all others of length 1 equals either 2n-5 or 2n-6, provided that n-r is not an odd integer.

2015-07-06math.AT

Moduli space of planar polygonal linkage: a combinatorial description

We explicitly describe a structure of a regular cell complex $K(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space $M$ is a sphere, the complex $K$ is dual to the boundary complex of the permutahedron. The dual complex $K^*$ is patched of Cartesian products of permutohedra and carries a natural PL-structure. It can be explicitly realized as a polyhedron in t…

2017-04-10math.AT

更多相关论文收起更多论文

On the middle dimensional homology classes of equilateral polygon spaces

Let $M_n$ be the configuration space of equilateral polygonal linkages with $n$ vertices in the Euclidean plane ${\mathbb R}^2$. We consider the case that $n$ is odd and set $n=2m+1$. In spite of the long history of research, the homology classes in $H_{m-1}(M_n;{\mathbb Z})$ are mysterious and not well-understood. Let $τ\colon M_n \to M_n$ be the involution induced by complex conjugation. In this paper, we determine the representation matrix of the homomorphism $τ_\ast\colon H_{m-1}(M_n;{\mathbb Z}) \to H_{m-1}(M_n;{\mathbb Z})$ with respect to a basis of $H_{m-1}(M_n;{\mathbb Z})$.

2015-07-11math.AGmath.AT

具有小遗传密码的平面多边形空间的更高拓扑复杂度

Higher topological complexity of planar polygon spaces having small genetic codes

本文将Davis关于小遗传密码平面多边形空间拓扑复杂度的界推广到更高拓扑复杂度，并在维数$m$为2的幂时确定其第$k$阶值为$km$或$km+1$。

2025-09-01math.AT

从射影辫子构型构建平面多边形空间

Building planar polygon spaces from the projective braid arrangement

本文证明平面多边形空间可通过沿由遗传码确定的子集合进行迭代细胞手术，从射影辫子构型构建，从而提供其紧化结构的新构造。

2024-02-05math.AT

On the zero-divisor-cup-length of spaces of oriented isometry classes of planar polygons

Using information about the rational cohomology ring of the space of oriented isometry classes of planar n-gons with specified side lengths, we obtain bounds for the zero-divisor-cup-length (zcl) of these spaces, which provide lower bounds for their topological complexity (TC). In many cases our result about the cohomology ring is complete and we determine the precise zcl. We find that there will usually be a significant gap between the bounds for TC implied by zcl and dimensional considerations.

2016-03-17math.AT

Real projective space as a space of planar polygons

We prove that real projective space RP^{n-3} is homeomorphic to the space of all isometry classes of n-gons in the plane with one side of length n-2 and all other sides of length 1. This makes the topological complexity of real projective space more relevant to robotics.

2015-01-16math.ATmath.GT

使用扩展持续同调变换进行平面对称性检测与量化

Planar Symmetry Detection and Quantification using the Extended Persistent Homology Transform

本文提出利用扩展持续同调变换将对称性表示为平滑实值量，以实现平面对象近似对称性的可视化、检测与量化，并分析其效果和局限。

2023-08-17math.AT

The isomorphism problem of planar polygon spaces

We give a proof of a Conjecture of Walker which states that one can recover the lengths of the bars of a circular linkage from the cohomology ring of the configuration space. For a large class of length vectors, this has been shown by Farber, Hausmann and Schuetz. In the remaining cases, we use Morse theory and the fundamental group to describe a subring of the cohomology invariant under graded ring isomorphism. From this subring the conjecture can be derived by applying a result of Gubeladze on the isomorphism problem of monoidal rings.

2009-06-24math.ATmath.GT

On the second homology of planar graph braid groups

We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.

2022-02-01math.ATmath.GT

Homology of planar telescopic linkages

We study topology of configuration spaces of planar linkages having one leg of variable length. Such telescopic legs are common in modern robotics where they are used for shock absorbtion and serve a variety of other purposes. Using a Morse theoretic technique, we compute explicitly, in terms of the metric data, the Betti numbers of configuration spaces of these mechanisms.

2009-09-16math.ATmath.DG

通过持续路径同调分析离散平面向量场中的奇异模式

Analyzing Singular Patterns in Discrete Planar Vector Fields via Persistent Path Homology

提出一种基于持续路径同调的方法，通过将离散平面向量场转换为有向图并计算其持续路径同调，有效分析含噪真实向量场的奇异模式位置与变化。

2025-07-16cs.CGmath.AT

Moduli Spaces of Planar Pentagonal Linkages: Combinatorial Description

Moduli spaces of planar polygonal linkages admit a cell structure which can be realized as a surgery on the permutohedron. We present a 3D visualization of the result of the surgery for all types of non-degenerate pentagonal linkages.

2013-05-30math.ATmath.GT

偶 Temperley-Lieb 代数与平面环的微分分次代数

Even Temperley-Lieb algebras and the dga of planar loops

本文证明偶数链Temperley-Lieb代数同调有丰富非平凡结构，其由平面环微分分次代数（dga）控制，还提供该dga的小型模型并据此推导同调结论。

2025-11-11math.ATmath.RT

等变K-同调与K-理论在一些离散平面仿射群中的应用

Equivariant K-homology and K-theory for some discrete planar affine groups

本文计算了三个离散平面仿射半直积群的Baum-Connes猜想两侧，即等变K-同调与约化群C*-代数的解析K-理论，利用3维模型和Bredon同调处理挠元，并为部分群提供了装配映射的显式生成元。

2023-11-26math.ATmath.GRmath.KT