5,311 research outputs found
A Note on Quasi-Triangulated Graphs
A graph is quasi-triangulated if each of its induced subgraphs has a vertex which is either simplicial (its neighbors form a clique) or cosimplicial (its nonneighbors form an independent set). We prove that a graph G is quasi-triangulated if and only if each induced subgraph H of G contains a vertex that does not lie in a hole, or an antihole, where a hole is a chordless cycle with at least four vertices, and an antihole is the complement of a hole. We also present an algorithm that recognizes a quasi-triangulated graph in O(nm) time
Triangulated surfaces in triangulated categories
For a triangulated category A with a 2-periodic dg-enhancement and a
triangulated oriented marked surface S we introduce a dg-category F(S,A)
parametrizing systems of exact triangles in A labelled by triangles of S. Our
main result is that F(S,A) is independent on the choice of a triangulation of S
up to essentially unique Morita equivalence. In particular, it admits a
canonical action of the mapping class group. The proof is based on general
properties of cyclic 2-Segal spaces.
In the simplest case, where A is the category of 2-periodic complexes of
vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya
category of the surface S. Therefore, our construction can be seen as
implementing a 2-dimensional instance of Kontsevich's program on localizing the
Fukaya category along a singular Lagrangian spine.Comment: 55 pages, v2: references added and typos corrected, v3: expanded
version, comments welcom
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
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