The Weber-Seifert dodecahedral space is non-Haken

@article{Burton2009TheWD,
  title={The Weber-Seifert dodecahedral space is non-Haken},
  author={Benjamin A. Burton and J. Hyam Rubinstein and Stephan Tillmann},
  journal={Transactions of the American Mathematical Society},
  year={2009},
  volume={364},
  pages={911-932}
}
In this paper we settle Thurston's old question of whether the Weber-Seifert dodecahedral space is non-Haken, a problem that has been a benchmark for progress in computational 3-manifold topology over recent decades. We resolve this question by combining recent significant advances in normal surface enumeration, new heuristic pruning techniques, and a new theoretical test that extends the seminal work of Jaco and Oertel. 
View PDF on arXiv
27 Citations
Highly Influential Citations
1
Background Citations
13
Methods Citations
6
Results Citations
1

27 Citations

Monopole Floer Homology, Eigenform Multiplicities, and the Seifert–Weber Dodecahedral Space

We show that the Seifert-Weber dodecahedral space $\mathsf{SW}$ is an $L$-space. The proof builds on our work relating Floer homology and spectral geometry of hyperbolic three-manifolds. A direct…

Incompressibility criteria for spun-normal surfaces

We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with non-empty boundary which has a quadrilateral in…

Computing closed essential surfaces in knot complements

A new, practical algorithm to test whether a knot complement contains a closed essential surface, derived from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory.

Virtual properties of 3-manifolds dedicated to the memory of Bill Thurston

We will discuss the proof of Waldhausen’s conjecture that compact aspherical 3-manifolds are virtually Haken, as well as Thurston’s conjecture that hyperbolic 3manifolds are virtually fibered. The…

The complexity of the normal surface solution space

A comprehensive analysis of millions of triangulations is undertaken and finds that in general the number of vertex normal surfaces is remarkably small, with strong evidence that the authors' pathological triangulation may in fact be the worst case scenarios.

A New Approach to Crushing 3-Manifold Triangulations

The aim of this paper is to make the crushing operation more accessible to practitioners, and easier to generalise to new settings, and to obtain a new practical and robust algorithm for non-orientable prime decomposition.

Subtle Transversality : a celebration of Allen Hatcher ’ s sixty-fifth birthday ( 09 w 2138 )

Allen Hatcher, probably most famous for his proof of the Smale Conjecture [6], has made significant contributions in a variety of areas centered on low-dimensional and geometric topology. Author of a…

Computational topology with Regina: Algorithms, heuristics and implementations

This paper documents for the first time in the literature some of the key algorithms, heuristics and implementations that are central to Regina's performance, and gives some historical background for the project.

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The least spanning area of a knot and the optimal bounding chain problem

It is shown here that a natural discrete version of the least area surface can be found in polynomial time, and it is proved that the related Optimal Homologous Chain Problem is NP-complete for homology with integer coefficients.

References

SHOWING 1-10 OF 25 REFERENCES

Converting between quadrilateral and standard solution sets in normal surface theory

A new algorithm for enumerating all standard vertex normal surfaces is obtained, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates.

P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst

A census of cusped hyperbolic 3-manifolds

The census contains descriptions of all hyperbolic 3-manifolds obtained by gluing the faces of at most seven ideal tetrahedra and various geometric and topological invariants are calculated for these manifolds.

NORMAL SURFACE Q-THEORY

We describe an approach to normal surface theory for triangulated 3-manifolds which uses only the quadrilateral disk types (Q-disks) to represent a nontrivial normal surface. Just as with regular…

The complexity of the normal surface solution space

A comprehensive analysis of millions of triangulations is undertaken and finds that in general the number of vertex normal surfaces is remarkably small, with strong evidence that the authors' pathological triangulation may in fact be the worst case scenarios.

Algorithms for essential surfaces in 3-manifolds

Several algorithms to find essential surfaces in 3dimensional manifolds are outlined, using normal and almost normal surface theory and the technique of crushing triangulations, to determine if a given 3-manifold has an embedded incompressible surface or not.

Isotopy classes of incompressible surfaces in irreducible 3-manifolds

Optimizing the double description method for normal surface enumeration

An account of the vertex enumeration problem as it applies to normal surfaces is given and new optimizations that yield strong improvements in both running time and memory consumption are presented.

Introducing Regina, The 3-Manifold Topology Software

Regina, a freely available software package for 3-manifold topologists, includes support for normal surfaces and angle structures and examples of research projects in which Regina has been used.

The maximum numbers of faces of a convex polytope

In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j d v , the maximum possible number of j -faces of a d -polytope with v…