9 research outputs found
On balanced planar graphs, following W. Thurston
Let be an orientation-preserving branched covering map of
degree , and let be an oriented Jordan curve passing through
the critical values of . Then is an oriented graph
on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced
balanced planar graphs and showed that they combinatorially characterize all
such , where has distinct critical values. We give a
detailed account of this discussion, along with some examples and an appendix
about Hurwitz numbers.Comment: 17 page
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
Brick polytopes of spherical subword complexes and generalized associahedra
International audienceWe generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra
K-Decompositions and 3d Gauge Theories
This paper combines several new constructions in mathematics and physics.
Mathematically, we study framed flat PGL(K,C)-connections on a large class of
3-manifolds M with boundary. We define a space L_K(M) of framed flat
connections on the boundary of M that extend to M. Our goal is to understand an
open part of L_K(M) as a Lagrangian in the symplectic space of framed flat
connections on the boundary, and as a K_2-Lagrangian, meaning that the
K_2-avatar of the symplectic form restricts to zero. We construct an open part
of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal
triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic
geometry, and combining them with the cluster coordinates for framed flat
PGL(K)-connections on surfaces. Using a canonical map from the complex of
configurations of decorated flags to the Bloch complex, we prove that any
generic component of L_K(M) is K_2-isotropic if the boundary satisfies some
topological constraints (Theorem 4.2). In some cases this implies that L_K(M)
is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier
on symplectic properties of PGL(2) gluing equations to reduce the
K_2-Lagrangian property to a combinatorial claim.
Physically, we use the symplectic properties of K-decompositions to construct
3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to
the compactification of K M5-branes on M. This extends known constructions for
K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of
abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead
to abelian mirror symmetries that are all generated by the elementary duality
between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence
that the degrees of freedom of T_K[M] grow cubically in K.Comment: 121 pages + 2 appendices, 80 figures; Version 2: reorganized
mathematical perspective, swapped Sections 3 and
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum