Configuration Spaces of Linkages in R^n

Universality theorems for linkages in the Minkowski plane

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of linkages in the Minkowski plane. We also give a proof of a differential universality theorem in the Minkowski plane: for any manifold M which is the interior of a compact manifold with boundary, there is a linkage which has a configuration space diffeomorphic to the disjoint uni…

Universality theorems for linkages in homogeneous surfaces

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of linkages in the (Lorentz-)Minkowski plane, in the hyperbolic plane and in the sphere. We also give a proof of a differential universality theorem in the Minkowski plane and in the hyperbolic plane: for any compact manifold M, there is a linkage whose configuration space is diffe…

On the Moduli Space of Multipolygonal Linkages in the Plane

The geometric, topological, and symplectic properties of moduli spaces (spaces of configurations modulo rotations and translations) of polygonal linkages have been studied by Kapovich, Millson, and Kamiyama, et. al. One can form a polygonal linkage by taking two free linkages and identifying initial and terminal vertices. This can be generalized so that one takes three free linkages and identifies initial and terminal vertices. Then one obtains a linkage which contains multiple polygons, any two of which have shared edges. The geometric and topological properties of moduli spaces of these mult…

Series Parallel Linkages

We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.

Extremal polygons in R^3

The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function $A$ in many cases is not a perfect Morse function. In particular, for an equilateral pentagonal linkage it has one extra local maximum (except for the global maximum) and one extra local minimum. In the present paper we consider the space of 3D configurations of a polygonal linkage. F…

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.

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.

Oriented area is a perfect Morse function

We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral polygons (which appear as Morse points) are interpreted as independent generators of the homology groups of the (decorated) configuration space.

平面蜘蛛构型空间上的平方距离函数及其在胡克能量与Voronoi距离中的应用

Squared Distance Function on the Configuration Space of a planar Spider with Applications to Hooke Energy and Voronoi Distance

本文针对平面蜘蛛机构构型空间尚未完全明晰的问题，利用平方距离函数的Morse理论开展研究，明确其一般为Morse-Bott函数，推导相关临界流形与指标公式，并将结果应用于胡克能量与Voronoi距离研究。