Oriented Flip Graphs and Noncrossing Tree Partitions

International audience Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.

Species with Potential Arising from Surfaces with Orbifold Points of Order 2, Part II: Arbitrary Weights

Let ${\boldsymbol{\Sigma }}=(\Sigma ,\mathbb{M},\mathbb{O})$ be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair $(\tau ,\omega )$ consisting of a triangulation $\tau $ of ${\boldsymbol{\Sigma }}$ and a function $\omega :\mathbb{O}\rightarrow \{1,4\}$, we define a chain complex $C_\bullet (\tau , \omega )$ with coefficients in $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$. Given ${\boldsymbol{\Sigma }}$ and $\omega $, we define a colored triangulation of ${\boldsymbol{\Sigma }_\omega }=(\Sigma ,\mathbb{M},\mathbb{O},\omega )$ to be a pair $(\tau ,\xi )$ consisting of a triangulation of ${\boldsymbol{\Sigma }}$ and a 1-cocycle in the cochain complex that is dual to $C_\bullet (\tau , \omega )$; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [25]. We define the flip graph of ${\boldsymbol{\Sigma }_\omega }$ as the graph whose vertices are the pairs $(\tau ,x)$ consisting of a triangulation $\tau $ and a cohomology class $x\in H^1(C^\bullet (\tau , \omega ))$, with an edge connecting two such pairs, $(\tau ,x)$ and $(\sigma ,z),$ if and only if there exist 1-cocycles $\xi \in x$ and $\zeta \in z$ such that $(\tau ,\xi )$ and $(\sigma ,\zeta )$ are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface $\Sigma $ is not contractible. In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class. We also give a full classification of the nondegenerate SPs one can associate to any given pair $(\tau ,\omega )$ over cyclic Galois extensions with certain roots of unity. The species constructed here are species realizations of the $2^{|\mathbb{O}|}$ skew-symmetrizable matrices that Felikson–Shapiro–Tumarkin associated in [17] to any given triangulation of ${\boldsymbol{\Sigma }}$. In the prequel [25] of this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures.

Cubical geometry in the polygonalisation complex

We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using properties of the flip graph, we show that the midcubes in the polygonalisation complex can be extended to a family of embedded and separating hyperplanes, parametrised by the arcs in the surface.We study the crossing graph of these hyperplanes and prove that it is quasi-isometric to the arc complex. We use the crossing graph to prove that, generically, different surfaces have different polygonalisation complexes. The polygonalisation complex is not CAT(0), but we can characterise the vertices where Gromov's link condition fails. This gives a tool for proving that, generically, the automorphism group of the polygonalisation complex is the (extended) mapping class group of the surface.

A Lower Bound on the Diameter of the Flip Graph

The flip graph is the graph whose vertices correspond to non-isomorphic combinatorial triangulations and whose edges connect pairs of triangulations that can be obtained from one another by flipping a single edge. In this note we show that the diameter of the flip graph is at least $\frac{7n}{3} + \Theta(1)$, improving upon the previous $2n + \Theta(1)$ lower bound.

EL-labelings and canonical spanning trees for subword complexes

International audience We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.