Homology of planar telescopic linkages

Telescopic Linkages and Topological Approach to Phase Transitions

A topological approach to the theory of equilibrium phase transitions in statistical physics is based on the Topological Hypothesis (TH), which claims that phase transitions are due to changes of the topology of suitable submanifolds in the configuration space. In this paper we examine in detail the anti-ferromagnetic mean-field XY model and study topology of the sub-energy manifolds. The latter can be interpreted mechanically as configuration space of a linkage with one telescopic leg. We use methods of Morse theory to describe explicitly the Betti numbers of this configuration space. We appl…

Homology of moduli spaces of linkages in high-dimensional Euclidean space

We study the topology of moduli spaces of closed linkages in \R^d depending on a length vector \ell\in \R^n. In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for odd d. In the case d=5 we calculate the Poincare polynomial in terms of combinatorial information encoded in the length vector.

Intersection homology of linkage spaces

We consider the moduli spaces $\mathcal{M}_d(\ell)$ of a closed linkage with n links and prescribed lengths in d-dimensional Euclidean space. For d>3 these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold. We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism type of the moduli spaces for a large class of length vectors in the case of d even. This result is a high-dimensional analogue of the Walker conjecture which was proven by Farber, Hausmann and the author.

Intersection homology of linkage spaces in odd dimensional Euclidean space

We consider the moduli spaces $\mathcal{M}_d(\ell)$ of a closed linkage with $n$ links and prescribed lengths $\ell\in \mathbb{R}^n$ in $d$-dimensional Euclidean space. For $d>3$ these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold. We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism types of $\mathcal{M}_d(\ell)$ for a large class of length vectors. These rings behave rather differently depending on whether $d$ is even or odd, with the even case having been treated in an earlier paper. The…

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 $H_{m-1}(M_n;{\mathbb Z})$.

机械臂的约束运动空间

Constrained Motion Spaces of Robotic Arms

该研究建立机械臂约束运动空间理论，在水平线、光滑曲线两种约束下计算其同调群，证明该计算等价于共线构型计数，依托Morse理论及相关连杆机构研究实现了拓扑到组合问题的转化。