134 research outputs found
Graphs of non-crossing perfect matchings
Let Pn be a set of n = 2m points that are the vertices of a convex polygon, and let Mm
be the graph having as vertices all the perfect matchings in the point set Pn whose edges
are straight line segments and do not cross, and edges joining two perfect matchings M1
and M2 if M2 = M1 ÂĄ (a; b) ÂĄ (c; d) + (a; d) + (b; c) for some points a; b; c; d of Pn. We
prove the following results about Mm: its diameter is m ÂĄ 1; it is bipartite for every m;
the connectivity is equal to m ÂĄ 1; it has no Hamilton path for m odd, m > 3; and finally
it has a Hamilton cycle for every m even, m>=4
Manifolds of isospectral arrow matrices
An arrow matrix is a matrix with zeroes outside the main diagonal, first row,
and first column. We consider the space of Hermitian arrow
-matrices with fixed simple spectrum . We prove
that this space is a smooth -manifold, and its smooth structure is
independent on the spectrum. Next, this manifold carries the locally standard
torus action: we describe the topology and combinatorics of its orbit space. If
, the orbit space is not a polytope, hence
this manifold is not quasitoric. However, there is a natural permutation action
on which induces the combined action of a semidirect product
. The orbit space of this large action is a simple
polytope. The structure of this polytope is described in the paper.
In case , the space is a solid torus with
boundary subdivided into hexagons in a regular way. This description allows to
compute the cohomology ring and equivariant cohomology ring of the
6-dimensional manifold using the general theory developed by
the first author. This theory is also applied to a certain -dimensional
manifold called the twin of . The twin carries a
half-dimensional torus action and has nontrivial tangent and normal bundles.Comment: 29 pages, 8 figure
Some notes on generic rectangulations
A rectangulation is a subdivision of a rectangle into rectangles. A generic rectangulation is a rectangulation that has no crossing segments. We explain several observations and pose some questions about generic rectangulations. In particular, we show how one may "centrally invert" a generic rectangulation about any given rectangle, analogous to reflection across a circle in classical geometry. We also explore 3-dimensional orthogonal polytopes related to "marked" rectangulations and drawings of planar maps. These observations arise from viewing a generic rectangulation as topologically equivalent to a sphere
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