November 18, 2010
It is important to have fast and effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work suggesting that for 3-sphere triangulations the reality is far different: we never need to add more than two tetrahedra, and we never need more than a handful of local modifications. If true in general, these extremely surprising results would have significant implications for decision algorithms and the study of triangulations in 3-manifold topology. The algorithms behind these experiments are interesting in their own right. Key techniques include the isomorph-free generation of all 3-manifold triangulations of a given size, polynomial-time computable signatures that identify triangulations uniquely up to isomorphism, and parallel algorithms for studying finite level sets in the infinite Pachner graph.
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October 27, 2011
It is important to have effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work that suggests the reality is far different: for an exhaustive census of 81,800,394 one-vertex triangulations that span 1,901 distinct closed orientable 3-manifolds,...
August 15, 2000
In this paper we describe a procedure to simplify any given triangulation of the 3-sphere using Pachner moves. We obtain an explicit exponential-type bound on the number of Pachner moves needed for this process. This leads to a new recognition algorithm for the 3-sphere.
March 11, 2023
We often rely on censuses of triangulations to guide our intuition in $3$-manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations; the current census only goes up to $10$ tetrahedra. Here, we show that it is feasible to search for large an...
May 30, 2016
Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the Pachner graph of such triangulations is connected. In this paper, we extend this result to show that (again with a small number of known exceptions), the subgraph...
October 11, 2023
We propose a Monte Carlo method to efficiently find, count, and sample abstract triangulations of a given manifold M. The method is based on a biased random walk through all possible triangulations of M (in the Pachner graph), constructed by combining (bi-stellar) moves with suitable chosen accept/reject probabilities (Metropolis-Hastings). Asymptotically, the method guarantees that samples of triangulations are drawn at random from a chosen probability. This enables us not o...
November 7, 2003
We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.
December 4, 2020
A key result in computational 3-manifold topology is that any two triangulations of the same 3-manifold are connected by a finite sequence of bistellar flips, also known as Pachner moves. One limitation of this result is that little is known about the structure of this sequences; knowing more about the structure could help both proofs and algorithms. Motivated by this, we show that there must be a sequence that satisfies a rigid property that we call "semi-monotonicity". We a...
February 6, 2019
We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes and bound on the lengths of edges of the triangulation. This leads to an algorithm to check from the combinatorics of the triangulation and bounds on lengths of edges, if two geometrically t...
December 5, 2014
A typical census of 3-manifolds contains all manifolds (under various constraints) that can be triangulated with at most n tetrahedra. Al- though 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 ...
June 19, 2005
In this survey on combinatorial properties of triangulated manifolds we discuss various lower bounds on the number of vertices of simplicial and combinatorial manifolds. Moreover, we give a list of all known examples of vertex-minimal triangulations.