The flip-graph of the 4-dimensional cube is connected

Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92 487 256 triangulations, partitioned into 247 451 symmetry classes.Comment: 20 pages, 3 figures, revised proofs and notation

An Edge-Based Framework for Enumerating 3-Manifold Triangulations

A typical census of 3-manifolds contains all manifolds (under various constraints) that can be triangulated with at most n tetrahedra. Although censuses are useful resources for mathematicians, constructing them is difficult: the best algorithms to date have not gone beyond n=12. The underlying algorithms essentially (i) enumerate all relevant 4-regular multigraphs on n nodes, and then (ii) for each multigraph G they enumerate possible 3-manifold triangulations with G as their dual 1-skeleton, of which there could be exponentially many. In practice, a small number of multigraphs often dominate the running times of census algorithms: for example, in a typical census on 10 tetrahedra, almost half of the running time is spent on just 0.3% of the graphs. Here we present a new algorithm for stage (ii), which is the computational bottleneck in this process. The key idea is to build triangulations by recursively constructing neighbourhoods of edges, in contrast to traditional algorithms which recursively glue together pairs of tetrahedron faces. We implement this algorithm, and find experimentally that whilst the overall performance is mixed, the new algorithm runs significantly faster on those "pathological" multigraphs for which existing methods are extremely slow. In this way the old and new algorithms complement one another, and together can yield significant performance improvements over either method alone

Face pairing graphs and 3-manifold enumeration

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal P^2-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal P^2-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the final theorem to the non-orientable case; v3: fixed a flaw in the proof of the conical face lemm

Equivelar and d-Covered Triangulations of Surfaces. I

We survey basic properties and bounds for $q$-equivelar and $d$-covered triangulations of closed surfaces. Included in the survey is a list of the known sources for $q$-equivelar and $d$-covered triangulations. We identify all orientable and non-orientable surfaces $M$ of Euler characteristic $0>\chi(M)\geq -230$ which admit non-neighborly $q$-equivelar triangulations with equality in the upper bound $q\leq\Bigl\lfloor\tfrac{1}{2}(5+\sqrt{49-24\chi (M)})\Bigl\rfloor$. These examples give rise to $d$-covered triangulations with equality in the upper bound $d\leq2\Bigl\lfloor\tfrac{1}{2}(5+\sqrt{49-24\chi (M)})\Bigl\rfloor$. A generalization of Ringel's cyclic $7{\rm mod}12$ series of neighborly orientable triangulations to a two-parameter family of cyclic orientable triangulations $R_{k,n}$, $k\geq 0$, $n\geq 7+12k$, is the main result of this paper. In particular, the two infinite subseries $R_{k,7+12k+1}$ and $R_{k,7+12k+2}$, $k\geq 1$, provide non-neighborly examples with equality for the upper bound for $q$ as well as derived examples with equality for the upper bound for $d$.Comment: 21 pages, 4 figure