Balanced Triangulations Research Articles

Centrally symmetric and balanced triangulations of [formula omitted] with few vertices

A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of Si×Sd−i−1 with 2d+2 vertices for all d≥3 and 1≤i≤d−2; they also proposed a balanced triangulation of S1×Sd−2 with 3d or 3d+2 vertices. In this paper, we provide another centrally symmetric (2d+2)-vertex triangulation of S2×Sd−3. We also construct the first balanced triangulation of S2×Sd−3 with 4d vertices, using a sphere decomposition inspired by handle theory.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.

Exceptional Balanced Triangulations on Surfaces

Izmestiev, Klee and Novik proved that any two balanced triangulations of a closed surface $$F^2$$ can be transformed into each other by a sequence of six operations called basic cross flips. Recently Murai and Suzuki proved that among these six operations only two operations are almost sufficient in the sense that, with for finitely many exceptions, any two balanced triangulations of a closed surface $$F^2$$ can be transformed into each other by these two operations. We investigate such finitely many exceptions, called exceptional balanced triangulations, and obtain the list of exceptional balanced triangulations of closed surfaces with low genera. Furthermore, we discuss the subsets $$\mathcal{O}$$ of the six operations satisfying the property that any two balanced triangulations of the same closed surface can be connected through a sequence of operations from $$\mathcal{O}$$.

A generalized lower bound theorem for balanced manifolds

A simplicial complex of dimension $$d-1$$ is said to be balanced if its graph is d-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if $$\Delta $$ is an arbitrary balanced triangulation of any closed homology manifold of dimension $$d-1 \ge 3$$ , then $$2h_2(\Delta ) - (d-1)h_1(\Delta ) \ge 4{d \atopwithdelims ()2}(\tilde{\beta }_1(\Delta )-\tilde{\beta }_0(\Delta ))$$ , thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag $$h''$$ -vectors.

Balanced subdivisions and flips on surfaces

In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2 2 -sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere.

Simplicial moves on balanced complexes

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly (d+1)-colored) triangulation of a combinatorial d-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial d-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every m≥d+2 and any closed combinatorial d-manifold M, two m-colored triangulations of M can be connected by a sequence of bistellar flips that preserve the vertex colorings.

Minimal Balanced Triangulations of Sphere Bundles over the Circle

We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb{S}^{d-2}$-bundles over $\mathbb{S}^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is nonorientable, the minimum number of vertices is $3d$; otherwise, it is $3d+2$. Similar results apply to all balanced simplicial complexes that triangulate homology manifolds with $\beta_1\neq0$ and $\beta_2=0$, where $\beta_i$'s are the Betti numbers, computed with coefficients in $\mathbb{Q}$.

Balanced triangulations

Motivated by applications in numerical analysis, we investigate balanced triangulations, i.e. triangulations where all angles are strictly larger than π/6 and strictly smaller than π/2, giving the optimal lower bound for the number of triangles in the case of the square. We also investigate platonic surfaces, where we find for each one its respective optimal bound. In particular, we settle (affirmatively) the open question whether there exist acute triangulations of the regular dodecahedral surface with 12 acute triangles [J. Itoh, T. Zamfirescu, Acute triangulations of the regular dodecahedral surface, European J. Combin. 28 (2007) 1072–1086].

LMT-skeleton heuristics for several new classes of optimal triangulations

Given a planar point set, we consider three classes of optimal triangulations: (1) the minimum weight triangulation with angular constraints (constraints on the minimum angle and the maximum angle in a triangulation), (2) the angular balanced triangulation which minimizes the sum of the ratios of the maximum angle to the minimum angle for each triangle, and (3) the area balanced triangulation which minimizes the variance of the areas of triangles in the triangulation. With appropriate definition of local optimality for each class, a simple unified method is established for the computation of the subgraphs of optimal triangulations. Computational experiments demonstrate that the method successfully identifies large portion of edges of the optimal triangulations of each class for all problem instances tested, and hence optimal triangulations for each class can be obtained from them by applying dynamic programming.