35,251 research outputs found
Polynomiality, Wall Crossings and Tropical Geometry of Rational Double Hurwitz Cycles
We study rational double Hurwitz cycles, i.e. loci of marked rational stable
curves admitting a map to the projective line with assigned ramification
profiles over two fixed branch points. Generalizing the phenomenon observed for
double Hurwitz numbers, such cycles are piecewise polynomial in the entries of
the special ramification; the chambers of polynomiality and wall crossings have
an explicit and "modular" description. A main goal of this paper is to
simultaneously carry out this investigation for the corresponding objects in
tropical geometry, underlining a precise combinatorial duality between
classical and tropical Hurwitz theory
Nested cycles in large triangulations and crossing-critical graphs
We show that every sufficiently large plane triangulation has a large
collection of nested cycles that either are pairwise disjoint, or pairwise
intersect in exactly one vertex, or pairwise intersect in exactly two vertices.
We apply this result to show that for each fixed positive integer , there
are only finitely many -crossing-critical simple graphs of average degree at
least six. Combined with the recent constructions of crossing-critical graphs
given by Bokal, this settles the question of for which numbers there is
an infinite family of -crossing-critical simple graphs of average degree
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
Study of the vacuum matrix element of products of parafields
We study the vacuum matrix elements of products of parafields using graphical
and combinatorial methods.Comment: 15 pages, 4 figures. Figures were omitted in the first versio
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