99 research outputs found
Treewidth, crushing, and hyperbolic volume
We prove that there exists a universal constant such that any closed
hyperbolic 3-manifold admits a triangulation of treewidth at most times its
volume. The converse is not true: we show there exists a sequence of hyperbolic
3-manifolds of bounded treewidth but volume approaching infinity. Along the
way, we prove that crushing a normal surface in a triangulation does not
increase the carving-width, and hence crushing any number of normal surfaces in
a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former
argument (in V1) used a construction that relied on a wrong theorem. Section
5.1 has also been adjusted to the new construction. Various other arguments
have been clarifie
Bounds for the genus of a normal surface
This paper gives sharp linear bounds on the genus of a normal surface in a
triangulated compact, orientable 3--manifold in terms of the quadrilaterals in
its cell decomposition---different bounds arise from varying hypotheses on the
surface or triangulation. Two applications of these bounds are given. First,
the minimal triangulations of the product of a closed surface and the closed
interval are determined. Second, an alternative approach to the realisation
problem using normal surface theory is shown to be less powerful than its dual
method using subcomplexes of polytopes.Comment: 38 pages, 25 figure
Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants
Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold.
In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds.
The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants
Finding large counterexamples by selectively exploring the Pachner graph
We often rely on censuses of triangulations to guide our intuition in
-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 tetrahedra. Here, we show that it is feasible to
search for large and hard-to-find counterexamples by using heuristics to
selectively (rather than exhaustively) enumerate triangulations. We use this
idea to find counterexamples to three conjectures which ask, for certain
-manifolds, whether one-vertex triangulations always have a "distinctive"
edge that would allow us to recognise the -manifold.Comment: 35 pages, 28 figures. A short version has been accepted for SoCG
2023; this full version contains some new results that do not appear in the
SoCG versio
On the complexity of cusped non-hyperbolicity
We show that the problem of showing that a cusped 3-manifold M is not hyperbolic is in NP, assuming S3-RECOGNITION is in coNP. To this end, we show that IRREDUCIBLE TOROIDAL RECOGNITION lies in NP. Along the way we unconditionally recover SATELLITE KNOT RECOGNITION lying in NP. This was previously known only assuming the Generalized Riemann Hypothesis. Our key contribution is to certify closed essential normal surfaces as essential in polynomial time in compact orientable irreducible ∂-irreducible triangulations. Our work is made possible by recent work of Lackenby showing several basic decision problems in 3-manifold topology are in NP or coNP.Mathematic
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