Related papers: Homology groups for particles on one-connected gra…
In this paper we study the homology and cohomology of confguration spaces of two distinct particles on a graph. Our main tool is intersection theory for cycles in graphs. We obtain an explicit description of the cohomology algebra of the…
The first and second homology groups are computed for configuration spaces of framed three-dimensional point particles with annihilation included, when up to two particles and an antiparticle are present.
In this thesis we develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to…
Using the ordered analogue of Farley-Sabalka's discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber's…
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell…
We study the ordered configuration spaces of star graphs. Inspired by the representation stability results of Church--Ellenberg--Farb for the ordered configuration space of a manifold and the edge stability results of…
We present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions,…
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…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we…
We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of…
We compute numerically the homology of several graph complexes in low loop orders, extending previous results.
We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…
In this paper we calculate the homology of configuration spaces of $n$ points on a circle, subject to the condition that two pre-determined points are included in the configuration. We make use of discrete Morse theory both to determine the…
We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data…
We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete…
The rational homology group of the order complex of non-even partitions of a finite set is calculated. A twisted version of the Goresky-MacPherson approach to similar homology calculations is proposed.
We present a new algorithm for computing the first discrete homology group of a graph. By testing the algorithm on different data sets of random graphs, we find that it significantly outperforms other known algorithms.
We investigate an algebraic problem related to the determination of the fundamental group of a class of spaces of configurations on surfaces. The configuration spaces are spaces of points grouped into colors. Whether two points are allowed…
Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…