Title page for ETD etd-0409103-153131


Type of Document Dissertation
Author Holcomb, Michael Edward
Author's Email Address holcomb@math.lsu.edu
URN etd-0409103-153131
Title On the Geometry and Topology of Moduli Spaces of Multi-Polygonal Linkages
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
J. William Hoffman Committee Chair
Dan Cohen Committee Member
James Madden Committee Member
Jorge Morales Committee Member
Larry Smolinsky Committee Member
Nicholas Apostolou Dean's Representative
Keywords
  • topology
  • moduli
  • linkages
Date of Defense 2003-03-31
Availability unrestricted
Abstract
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 multi-polygonal linkages are studied. These spaces turn out to be compact algebraic varieties. Multi-quadrilateral linkages whose moduli spaces are at most one dimensional are classified. The dimensions and some Euler characteristics are computed, and conditions under which these spaces are smooth manifolds are determined. Some conditions are also given for when the moduli spaces are connected and when they are disjoint unions of two moduli spaces of polygonal linkages.
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