1,075 research outputs found
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
This paper is a concise introduction to virtual knot theory, coupled with a
list of research problems in this field.Comment: 65 pages, 24 figures. arXiv admin note: text overlap with
arXiv:math/040542
The Euler anomaly and scale factors in Liouville/Toda CFTs
The role played by the Euler anomaly in the dictionary relating sphere
partition functions of four dimensional theories of class and two
dimensional nonrational CFTs is clarified. On the two dimensional side, this
involves a careful treatment of scale factors in Liouville/Toda correlators.
Using ideas from tinkertoy constructions for Gaiotto duality, a framework is
proposed for evaluating these scale factors. The representation theory of Weyl
groups plays a critical role in this framework.Comment: 55 pages, 16 figures; v2:fixed referencing & typos ; v3: argument
about scale factors in Liouville/Toda now phrased in terms of stripped
correlators, leading to a sharper conjecture (earlier version had some
inaccurate statements). Presentation improved, typos fixed, refs added. I
thank the anonymous referee for comments. Version accepted for publication in
JHE
Stable Degenerations of Surfaces Isogenous to a Product II
In this note, we describe the possible singularities on a stable surface
which is in the boundary of the moduli space of surfaces isogenous to a
product. Then we use the -Gorenstein deformation theory to get some
connected components of the moduli space of stable surfaces.Comment: 17 pages; the preliminary part is made more concise. Accecpted by
Transactions of the American Mathematical Societ
Degenerate crossing number and signed reversal distance
The degenerate crossing number of a graph is the minimum number of transverse
crossings among all its drawings, where edges are represented as simple arcs
and multiple edges passing through the same point are counted as a single
crossing. Interpreting each crossing as a cross-cap induces an embedding into a
non-orientable surface. In 2007, Mohar showed that the degenerate crossing
number of a graph is at most its non-orientable genus and he conjectured that
these quantities are equal for every graph. He also made the stronger
conjecture that this also holds for any loopless pseudotriangulation with a
fixed embedding scheme.
In this paper, we prove a structure theorem that almost completely classifies
the loopless 2-vertex embedding schemes for which the degenerate crossing
number equals the non-orientable genus. In particular, we provide a
counterexample to Mohar's stronger conjecture, but show that in the vast
majority of the 2-vertex cases, the conjecture does hold.
The reversal distance between two signed permutations is the minimum number
of reversals that transform one permutation to the other one. If we represent
the trajectory of each element of a signed permutation under successive
reversals by a simple arc, we obtain a drawing of a 2-vertex embedding scheme
with degenerate crossings. Our main result is proved by leveraging this
connection and a classical result in genome rearrangement (the
Hannenhali-Pevzner algorithm) and can also be understood as an extension of
this algorithm when the reversals do not necessarily happen in a monotone
order.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Dimer Models from Mirror Symmetry and Quivering Amoebae
Dimer models are 2-dimensional combinatorial systems that have been shown to
encode the gauge groups, matter content and tree-level superpotential of the
world-volume quiver gauge theories obtained by placing D3-branes at the tip of
a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the
quiver graph. However, the string theoretic explanation of this was unclear. In
this paper we use mirror symmetry to shed light on this: the dimer models live
on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is
wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the
singular point, and geometrically encode the same quiver theory on their
world-volume.Comment: 55 pages, 27 figures, LaTeX2
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