Topology Of Configuration Space Research Articles

Controlling the configuration space topology of mechanical structures.

Linkages are mechanical devices constructed from rigid bars and freely rotating joints studied both for their utility in engineering and as mathematical idealizations in a number of physical systems. Recently, there has been a resurgence of interest in designing linkages in the physics community due to the concurrent developments of mechanical metamaterials, topological mechanics, and the discovery of anomalous rigidity in fiber networks and vertex models. These developments raise a natural question: to what extent can the motion of a linkage or mechanical structure be designed? Here, we describe a method to design the topology of the configuration space of a linkage by first identifying the manifold of critical points, then perturbing around such critical configurations. Unlike other methods, our methods are tractable and provide a simple visual toolkit for mechanism design. We demonstrate our procedure by designing a mechanism to gate the propagation of a soliton in a Kane-Lubensky chain of interconnected rotors.

Интегрируемые системы в планарной робототехнике

Основная цель данной статьи — исследовать коммутирующие потоки и интегрируемые системы на конфигурационом пространстве плоских шарнирных многоугольников. Наше исследование приводит к определению естественной формы объема на каждом конфигурационном пространтве плоских шарнирных многоугольников, понятию перекрестных произведений интегрируемых систем, а также к понятию мульти-намбу интегрируемых систем. Первые интегралы наших систем являются функциями типа Ботта-Морса, которые могут быть использованы для изучения топологии конфигурационных пространств.

Anyons from three-body hard-core interactions in one dimension

Traditional anyons in two dimensions have generalized exchange statistics governed by the braid group. By analyzing the topology of configuration space, we discover that an alternate generalization of the symmetric group governs particle exchanges when there are hard-core three-body interactions in one-dimension. We call this new exchange symmetry the traid group and demonstrate that it has abelian and non-abelian representations that are neither bosonic nor fermionic, and which also transform differently under particle exchanges than braid group anyons. We show that generalized exchange statistics occur because, like hard-core two-body interactions in two dimensions, hard-core three-body interactions in one dimension create defects with co-dimension two that make configuration space no longer simply-connected. Ultracold atoms in effectively one-dimensional optical traps provide a possible implementation for this alternate manifestation of anyonic physics.

Topology of configuration space of the mean-field ϕ^{4} model by Morse theory.

In this paper we present the study of the topology of the equipotential hypersurfaces of configuration space of the mean-field ϕ^{4} model with a Z_{2} symmetry. Our purpose is discovering, if any, the relation between the second-order Z_{2}-symmetry-breaking phase transition and the geometric entities mentioned above. The mean-field interaction allows us to solve analytically either the thermodynamic in the canonical ensemble or the topology by means of Morse theory. We have analyzed the results in the light of two theorems on sufficiency conditions for symmetry-breaking phase transitions recently proven. This study is part of a research line based on the general framework of geometric-topological approach to Hamiltonian chaos and critical phenomena.

Coincidence Structures and Hard-Core Few-Body Interactions

The symmetry and topology of the coincidence structure, i.e. the locus of points in configuration space corresponding to particles in the same position, plays a critical role in extracting universal properties for few-body models with hard-core interactions. The coincidence structure is a scale-invariant union of manifolds possessing rich symmetry. When there are zero-range hard-core two-body interactions, the coincidence structure forms a nodal surface for finite-energy wave functions in configuration space. More generally, it acts like a defect that changes the topology of configuration space in a way that depends on the dimension of the underlying space, the total number of particles, and the number of particles in the hard-core interaction. We show that for the specific case of three-body hard-core interactions in one-dimension, the configuration space is no longer simply-connected, providing a topological explanation for several models that exhibit anyonic behavior.

Homology groups for particles on one-connected graphs

We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our approach is based on some fundamental combinatorial properties of the configuration spaces, Mayer-Vietoris sequences for different parts of configuration spaces, and some limited use of discrete Morse theory. As one of the results, we derive the closed-form formulae for ranks of the homology groups for indistinguishable particles on tree graphs. We also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles. Our motivation is the search for new kinds of quantum statistics.

New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations

We use topology of configuration spaces to give a characterization of Neuwirth--Stallings pairs $(S^5, K)$ with $\dim K = 2$. As a consequence, we construct polynomial map germs $(\mathbb{R}^{6},0)\to (\mathbb{R}^{3},0)$ with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a disk, thus putting an end to Milnor's non-triviality question. Furthermore, for a polynomial map germ $(\mathbb{R}^{2n},0) \to (\mathbb{R}^{n},0)$ or $(\mathbb{R}^{2n+1},0) \to (\mathbb{R}^{n},0)$, $n \geq 3$, with an isolated singularity at the origin, we study the conditions under which the associated Milnor fiber has the homotopy type of a bouquet of spheres. We then construct, for every pair $(n, p)$ with $n/2 \geq p \geq 2$, a new example of a polynomial map germ $(\mathbb{R}^n,0) \to (\mathbb{R}^p,0)$ with an isolated singularity at the origin such that its Milnor fiber has the homotopy type of a bouquet of a positive number of spheres.

The Euler characteristic of the configuration space of planar spidery linkages

The topology of configuration spaces of planar mechanical linkages has been studied by many people. Among these linkages, polygonal and spidery ones are particularly important. We call the configuration space of a polygonal linkage a polygon space. The study of polygon spaces was originated by Hausmann [7], Kapovich and Millson [10] and Walker [15]. Their study has a long history and has begotten many references. For example, the homology groups of polygon spaces of arbitrary edge lengths have been determined by Farber and Schutz [6].

Configuration spaces of convex and embedded polygons in the plane

This paper concerns the topology of configuration spaces of linkages whose underlying graph is a single cycle. Assume that the edge lengths are such that there are no configurations in which all the edges lie along a line. The main results are that, modulo translations and rotations, each component of the space of convex configurations is homeomorphic to a closed Euclidean ball and each component of the space of embedded configurations is homeomorphic to a Euclidean space. This represents an elaboration on the topological information that follows from the convexification theorem of Connelly, Demaine, and Rote.

Enhanced quantization on the circle

We apply the quantization scheme introduced in Klauder 2012 (arXiv:1204.2870) to a particle on a circle. We find that the quantum action functional restricted to appropriate coherent states can be expressed as the classical action plus ℏ-corrections. We succeed in proving this for a Hamiltonian endowed with a general periodic potential. This result extends the examples presented in the cited paper for a system on a nontrivial configuration space topology.

Computational topology for configuration spaces of hard disks

We explore the topology of configuration spaces of hard disks experimentally and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik, and Kahle that critical points correspond to configurations of disks with balanced mechanical stresses and suggest conjectures about the asymptotic topology as the number of disks tends to infinity.

Arbeitsgemeinschaft: Topological Robotics

The purpose of the Arbeitsgemeinschaft was to enable PhD students and researchers to study Topological Robotics, a new field investigating topological problems motivated by robotics and engineering as well as problems of practical robotics requiring topological tools. The topics broadly fell into the areas of Topology of configuration spaces, Topological complexity of robot motion planning algorithms and Stochastic topology.

Topology of configuration space of two particles on a graph, II

This paper continues the investigation of the configuration space of two distinct points on a graph. We analyze the process of adding an additional edge to the graph and the resulting changes in the topology of the configuration space. We introduce a linking bilinear form on the homology group of the graph with values in the cokernel of the intersection form (introduced in Part I of this work). For a large class of graphs, which we call mature graphs, we give explicit expressions for the homology groups of the configuration space. We show that under a simple condition, adding an edge to a mature graph yields another mature graph.

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.

Orbit configuration spaces associated to discrete subgroups of [formula omitted

The purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from a group G acting freely, and properly discontinuously on the upper half-plane H 2 . The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to 1. the Lie algebra obtained from the higher homotopy groups of analogous constructions associated to H 2 × C q modulo torsion, as well as 2. the Lie algebra obtained from horizontal chord diagrams for surfaces. The resulting Lie algebras are similar to those studied in [T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math. 78 (1988) 339–363; T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math. 179 (1994) 123–138; T. Kohno, Elliptic KZ system, braid groups of the torus and Vassiliev invariants, Topology and its Applications 78 (1997) 79–94; D.C. Cohen, Monodromy of fiber-type arrangements and orbit configuration spaces, Forum Math. 13 (2001) 505–530; F.R. Cohen, M. Xicoténcatl, On orbit configuration spaces associated to the Gaussian integers: homology and homotopy groups, Topology Appl. 118 (2002) 17–29; E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. J. Julius Schauder Center 11 (1998), 249–271; E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, in: Springer Monographs in Mathematics, Springer-Verlag, 2001; F.R. Cohen and T. Sato, On groups of morphisms of coalgebras, (submitted for publication)]. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by G is also addressed.

Observable topological effects in molecular devices with Möbius topology

We study the topological properties of quantum states for the spinless particle hopping in a M\"obius ladder. This system can be regarded as a molecular device possibly engineered from the M\"obius annulenes, which enjoys a pseudospin orbital interaction described by a non-Abelian gauge structure. It results from the nontrivial topology of configuration space and results in various observable effects, such as optical spectral splitting. The transmission spectrum through the M\"obius molecular device is calculated to demonstrate a topological effect as destructive interferences in the conduction band. The induced interaction also leads to an entanglement between the transverse and longitudinal modes for any locally factorized state.

Topology of configuration space of two particles on a graph, I

In this paper we study the homology and cohomology of configuration spaces F(Γ,2) of two distinct particles on a graph Γ. Our main tool is intersection theory for cycles in graphs. We obtain an explicit description of the cohomology algebra H∗(F(Γ,2);Q) in the case of planar graphs.

Phase transitions and configuration space topology

Equilibrium phase transitions may be defined as nonanalytic points of thermodynamic functions, e.g., of the canonical free energy. Given a certain physical system, it is of interest to understand which properties of the system account for the presence, or the absence, of a phase transition, and an investigation of these properties may lead to a deeper understanding of the physical phenomenon. One possible way to approach this problem, reviewed and discussed in the present paper, is the study of topology changes in configuration space which are found to be related to equilibrium phase transitions in classical statistical mechanical systems. For the study of configuration space topology, one considers the subsets ${\mathcal{M}}_{v}$, consisting of all points from configuration space with a potential energy per particle equal to or less than a given $v$. For finite systems, topology changes of ${\mathcal{M}}_{v}$ are intimately related to nonanalytic points of the microcanonical entropy. In the thermodynamic limit, a more complex relation between nonanalytic points of thermodynamic functions (i.e., phase transitions) and topology changes is observed. For some class of short-range systems, a topology change of the ${\mathcal{M}}_{v}$ at $v={v}_{t}$ was proven to be necessary, but not sufficient, for a phase transition to take place at a potential energy ${v}_{t}$. In contrast, phase transitions in systems with long-range interactions or in systems with nonconfining potentials need not be accompanied by such a topology change. Instead, for such systems the nonanalytic point in a thermodynamic function is found to have some maximization procedure at its origin. These results may foster insight into the mechanisms which lead to the occurrence of a phase transition, and thus may help to explore the origin of this physical phenomenon.

Fermionic Quantization and Configuration Spaces for the Skyrme and Faddeev-Hopf Models

The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is S 2. It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given.

The mean-fieldφ4model: Entropy, analyticity, and configuration space topology

A large deviation technique is applied to the mean-field model Phi4, providing an exact expression for the configurational entropy s(v,m) as a function of the potential energy v and the magnetization m. Although a continuous phase transition occurs at some critical energy vc, the entropy is found to be a real analytic function in both arguments, and it is only the maximization over m which gives rise to a nonanalyticity in s(v)=supm s(v,m). This mechanism of nonanalyticity-generation by maximization over one variable of a real analytic entropy function is restricted to systems with long-range interactions and has--for continuous phase transitions--the generic occurrence of classical critical exponents as an immediate consequence. Furthermore, this mechanism can provide an explanation why, contradictory to the so-called topological hypothesis, the phase transition in the mean-field model need not be accompanied by a topology change in the family of constant-energy submanifolds.