759 research outputs found
Symplectic fillings of Seifert fibered spaces
We give finiteness results and some classifications up to diffeomorphism of
minimal strong symplectic fillings of Seifert fibered spaces over S^2
satisfying certain conditions, with a fixed natural contact structure. In some
cases we can prove that all symplectic fillings are obtained by rational
blow-downs of a plumbing of spheres. In other cases, we produce new manifolds
with convex symplectic boundary, thus yielding new cut-and-paste operations on
symplectic manifolds containing certain configurations of symplectic spheres.Comment: 39 pages, 21 figures, v2 a few minor corrections and citations, v3
added clarifications in the proof of Lemma 2.8, plus some minor change
Embedding of metric graphs on hyperbolic surfaces
An embedding of a metric graph on a closed hyperbolic surface is
\emph{essential}, if each complementary region has a negative Euler
characteristic. We show, by construction, that given any metric graph, its
metric can be rescaled so that it admits an essential and isometric embedding
on a closed hyperbolic surface. The essential genus of is the
lowest genus of a surface on which such an embedding is possible. In the next
result, we establish a formula to compute . Furthermore, we show that
for every integer , admits such an embedding (possibly
after a rescaling of ) on a surface of genus .
Next, we study minimal embeddings where each complementary region has Euler
characteristic . The maximum essential genus of is
the largest genus of a surface on which the graph is minimally embedded.
Finally, we describe a method explicitly for an essential embedding of , where and are realized.Comment: Revised version, 11 pages, 3 figure
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
Surfaces with given Automorphism Group
Frucht showed that, for any finite group , there exists a cubic graph such
that its automorphism group is isomorphic to . For groups generated by two
elements we simplify his construction to a graph with fewer nodes. In the
general case, we address an oversight in Frucht's construction. We prove the
existence of cycle double covers of the resulting graphs, leading to simplicial
surfaces with given automorphism group. For almost all finite non-abelian
simple groups we give alternative constructions based on graphic regular
representations. In the general cases for , we provide
alternative constructions of simplicial spheres. Furthermore, we embed these
surfaces into the Euclidean 3-Space with equilateral triangles such that the
automorphism group of the surface and the symmetry group of the corresponding
polyhedron in are isomorphic
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