Counting essential surfaces in 3-manifolds

@article{Dunfield2020CountingES,
  title={Counting essential surfaces in 3-manifolds},
  author={Nathan M. Dunfield and Stavros Garoufalidis and J. Hyam Rubinstein},
  journal={Inventiones mathematicae},
  year={2020},
  volume={228},
  pages={717 - 775}
}
We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give… 

Essential Surfaces in the Exterior of K13n586

We count the number of isotopy classes of closed, connected, orientable, essential surfaces embedded in the exteriorB of the knot K13n586. The main result is that the count of surfaces by genus is

The Number of Closed Essential Surfaces in Montesinos Knots with Four Rational Tangles

. In the complement of a hyperbolic Montesinos knot with 4 rational tangles, we investigate the number of closed, connected, essential, orientable surfaces of a fixed genus g , up to isotopy. We show

Infinitely Many Knots With NonIntegral Trace

We prove that there are infinitely many non-homeomorphic hyperbolic knot complements $S^3\setminus K_i = \mathbb{H}^3/\Gamma_i$ for which $\Gamma_i$ contains elements whose trace is an algebraic

References

SHOWING 1-10 OF 74 REFERENCES

Computing closed essential surfaces in 3-manifolds

We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and

Essential Surfaces in the Exterior of K13n586

We count the number of isotopy classes of closed, connected, orientable, essential surfaces embedded in the exteriorB of the knot K13n586. The main result is that the count of surfaces by genus is

Measured Lamination Spaces for 3-Manifolds

One of Thurston’s important contributions to surface theory is his construction of measured lamination spaces. These spaces originally arose because their projectivizations form natural boundaries

Quadrilateral–Octagon Coordinates for Almost Normal Surfaces

Joint coordinates is introduced, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties and can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein, and Thompson, reducing experimental running times by factors of thousands.

Degenerations of hyperbolic structures, II: Measured laminations in 3-manifolds

That paper concerned the general theory of groups acting on R-trees and the relationship of these actions to representations into SL2(C). The purpose of the present paper is to develop the

Incompressible surfaces via branched surfaces

Normal surfaces in topologically finite 3-manifolds

The concept of a normal surface in a triangulated, compact 3-manifold was generalised by Thurston to a spun-normal surface in a non-compact 3-manifold with ideal triangulation. This paper defines a

The cusped hyperbolic census is complete

For the first time, it is proved here that the census meets its aim: it is rigorously certify that every ideal 3-manifold triangulation with <= 8 tetrahedra is either homeomorphic to one of the census manifolds, or non-hyperbolic.

Triangulations of hyperbolic 3‐manifolds admitting strict angle structures

It is conjectured that every cusped hyperbolic 3‐manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a ‘geometric’ triangulation of the manifold). Under a mild homology

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.
...