1,125 research outputs found
The genus of curve, pants and flip graphs
This article is about the graph genus of certain well studied graphs in
surface theory: the curve, pants and flip graphs. We study both the genus of
these graphs and the genus of their quotients by the mapping class group. The
full graphs, except for in some low complexity cases, all have infinite genus.
The curve graph once quotiented by the mapping class group has the genus of a
complete graph so its genus is well known by a theorem of Ringel and Youngs.
For the other two graphs we are able to identify the precise growth rate of the
graph genus in terms of the genus of the underlying surface. The lower bounds
are shown using probabilistic methods.Comment: 26 pages, 9 figure
Random super matrices with an external source
In the past we have considered Gaussian random matrix ensembles in the
presence of an external matrix source. The reason was that it allowed, through
an appropriate tuning of the eigenvalues of the source, to obtain results on
non-trivial dual models, such as Kontsevich's Airy matrix models and
generalizations. The techniques relied on explicit computations of the k-point
functions for arbitrary N (the size of the matrices) and on an N-k duality.
Numerous results on the intersection numbers of the moduli space of curves were
obtained by this technique. In order to generalize these results to include
surfaces with boundaries, we have extended these techniques to supermatrices.
Again we have obtained quite remarkable explicit expressions for the k-point
functions, as well as a duality. Although supermatrix models a priori lead to
the same matrix models of 2d-gravity, the external source extensions considered
in this article lead to new geometric results.Comment: 12 page
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
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