Skip to main content
Article thumbnail
Location of Repository

Configuration spaces of thick particles on graphs



In this thesis, we study the topology of configuration spaces of particles of variable radius r>0 moving on a metric graph. Our main tool is a piecewise linear (PL) Morse-Bott theory for affine polytope complexes, which extends the Morse theory for such complexes introduced by M. Bestvina and N. Brady.\ud As the size parameter r increases, the topological properties of the corresponding configuration spaces vary. We show that there are finitely many critical values where the homotopy type of these spaces changes, and describe these critical values in terms of metric properties of the graph. This provides an upper bound on the number of critical values in terms of the metric data. Moreover, we apply PL Morse-Bott theory to analyse the change in homotopy type of configuration spaces of thick particles when the radius transits a critical value.\ud Provided that r is sufficiently small, we show that the thick particle configuration space is homotopy equivalent to the familiar configuration space of zero-size points on the graph. We also investigate discrete models for configuration spaces of two thick particles. Moreover, given a metric graph and the size parameter r, we provide an algorithm for computing the number of path-components of the configuration space of two thick particles

Year: 2011
OAI identifier:
Provided by: Durham e-Theses

Suggested articles


  1. (2001). A course in metric geometry, doi
  2. (2009). Abrams's stable equivalence for graph braid groups, doi
  3. (1988). Advanced Engineering Mathematics, Sixth Edition,
  4. (2002). Algebraic topology, doi
  5. (1967). Algebraic topology: an introduction, Harcourt, Brace and World,
  6. and Thesaurus (2007), Edited by Maurice Waite,
  7. (2011). Automated guided vehicle (accessed
  8. (2004). Braid groups and right angled Artin groups, doi
  9. (2005). Braid groups of the sun graph,
  10. (2005). Collision free motion planning on graphs, in: \Algorithmic Foundations of Robotics IV", doi
  11. (2009). Computing braid groups of graphs with applications to robot motion planning, to appear in Homology, Homotopy and Applications. Available at: doi
  12. (2000). Con spaces and braid groups of graphs,
  13. (2001). Con spaces and braid groups on graphs in robotics,
  14. (2002). Con spaces of colored graphs,
  15. (2011). Con spaces of thick particles on a metric graph, accepted on 26/4/2011, to appear in Algebraic and Geometric Topology. doi
  16. (2002). Con spaces of weighted graphs in high-dimensional Euclidean spaces, Contributions to Algebra and Geometry,
  17. (2007). Con spaces, braids and robotics, survey for summer school on braids & applications at the National University of Singapore. Available at: singaporetutorial.pdf.
  18. (2010). Dijkstra's algorithm (accessed doi
  19. (2005). Discrete Morse theory and graph braid groups, doi
  20. (2010). Discretized con and partial partitions, doi
  21. (2001). Euler characteristic of the con space of a complex, doi
  22. (2002). Finding topology in a factory: con spaces, doi
  23. (2009). Graph topologies induced by edge lengths, doi
  24. Handbook of Robotics doi
  25. (1965). Homology of deleted products in dimension one, doi
  26. (1970). Homology of deleted products of one-dimensional spaces, doi
  27. (2006). Homology of tree braid groups, in: \Topological and Asymptotic Aspects of Group Theory", doi
  28. (1961). Homotopy groups of certain deleted product spaces, doi
  29. Hyoung Ko and Hyo Won Park (2011), Characteristics of graph braid groups, doi
  30. (2008). Industrial Robotics, in: \Springer Handbook of Robotics" doi
  31. (2004). Introduction to autonomous mobile robots, doi
  32. (2008). Invitation to topological robotics, doi
  33. Ko cak (2009), Stability of graphs,
  34. (2005). Metrized graphs, electrical networks, and Fourier analysis, doi
  35. (1997). Morse theory and properties of groups, doi
  36. (1998). Morse theory for cell complexes, doi
  37. (2008). On the cohomology rings of tree braid groups, doi
  38. (1941). Ordered topological spaces, doi
  39. (2007). Presentations for the cohomology rings of tree braid groups, in: \Topology and Robotics", doi
  40. (2009). Presentations of graph braid groups, doi
  41. (2011). Robot (accessed
  42. (1998). Safe cooperative robotic motions via dynamics on graphs, doi
  43. (2002). Safe, cooperative robot dynamics on graphs, doi
  44. (2006). Scienti methods in mobile robotics: quantitative analysis of agent behaviour, Springer series in advanced manufacturing, Springer{Verlag London Limited.
  45. (2010). Sparse stable packings of hard discs in a box,
  46. Swi atkowski (2001), Estimates for homological dimension of con spaces of graphs, doi
  47. (2009). The con space of two particles moving on a graph, PhD thesis,
  48. (1962). The fundamental group of certain deleted product spaces, doi
  49. (2009). The Markov chain Monte Carlo revolution, doi
  50. (2007). Topological complexity of collision-free motion planning algorithms in the presence of multiple moving obstacles, \Topology and Robotics" doi
  51. (2009). Topology of con space of two particles on a graph, I, Algebraic and Geometric Topology, doi
  52. (2010). Topology of con space of two particles on a graph, II, Algebraic and Geometric Topology, doi
  53. (2010). Uniqueness of electrical currents in a network of nite total resistance, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.