Asymptotically efficient triangulations of the d-cube

Let $P$ and $Q$ be polytopes, the first of "low" dimension and the second of "high" dimension. We show how to triangulate the product $P \times Q$ efficiently (i.e., with few simplices) starting with a given triangulation of $Q$. Our method has a computational part, where we need to compute an efficient triangulation of $P \times \Delta^m$, for a (small) natural number $m$ of our choice. $\Delta^m$ denotes the $m$-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube $I^n$: We decompose $I^n = I^k \times I^{n-k}$, for a small $k$. Then we recursively assume we have obtained an efficient triangulation of the second factor and use our method to triangulate the product. The outcome is that using $k=3$ and $m=2$, we can triangulate $I^n$ with $O(0.816^{n} n!)$ simplices, instead of the $O(0.840^{n} n!)$ achievable before.Comment: 19 pages, 6 figures. Only minor changes from previous versions, some suggested by anonymous referees. Paper accepted in "Discrete and Computational Geometry

Lattice Delone simplices with super-exponential volume

In this short note we give a construction of an infinite series of Delone simplices whose relative volume grows super-exponentially with their dimension. This dramatically improves the previous best lower bound, which was linear.Comment: 7 pages; v2: revised version improves our exponential lower bound to a super-exponential on

There are only two nonobtuse binary triangulations of the unit nn-cube

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into $n!$ simplices. In this paper we show that, surprisingly, for each $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than $n!({\rm e}-2)$.Comment: 17 pages, 7 figure

Lower bounds for the simplexity of the n-cube

In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of $n$-dimensional cubes. In particular we show that the number of simplices in dissections of $n$-cubes without additional vertices is at least $(n+1)^{\frac {n-1} 2}$.Comment: 10 page

Extremal properties for dissections of convex 3-polytopes

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.Comment: 19 page

Fixed parameter tractable algorithms in combinatorial topology

To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an exponential-time enumeration. However, asymptotically most graphs do not result in any 3-manifold triangulation, which leads to significant "wasted time" in topological enumeration algorithms. Here we give a new algorithm to determine whether a given face pairing graph supports any 3-manifold triangulation, and show this to be fixed parameter tractable in the treewidth of the graph. We extend this result to a "meta-theorem" by defining a broad class of properties of triangulations, each with a corresponding fixed parameter tractable existence algorithm. We explicitly implement this algorithm in the most generic setting, and we identify heuristics that in practice are seen to mitigate the large constants that so often occur in parameterised complexity, highlighting the practicality of our techniques.Comment: 16 pages, 9 figure