Low Dimensional Topology

November 19, 2007

Hyperbolic geometry and normal surfaces

(Reprinted from my old ldt blog)

I’ve mentioned before the idea of WYSIWYG (what you see is what you get) topology, the idea that we should be able to use the combinatorial data in structures like trianglulations and Heegaard splittings in order to understand the (hyperbolic) geometry of the ambient manifold. Mark Lackenby’s recent preprint [1] does the converse, using the geometry of a hyperbolic knot complement to show that there is an algorithm to calculate its tunnel number (aka Heegaard genus minus one). The algorithm is based on normal surface theory, which has been around for a while, but uses the hyperbolic geometry to fix the main problem with normal surface theory: normal tori.

A normal or almost normal surface is one that intersects a triangulation in a simple way that allows one to write the surface as an integral solution to a certain collection of linear equations. Because triangulations adapt well to algorithms, many things about normal and almost normal surfaces can be calculated (for example by using the program Regina). Haken showed [2] that every incompressible surface is isotopic to a normal surface and used this to prove some finiteness results. More recently, Stocking [3] and Rubinstein [4] showed independently that every strongly irreducible Heegaard surface is isotopic to an almost normal surface. Thus to study the Heegaard splittings of a 3-manifold (in particular to calculate the Heegaard genus/tunnel number), one can try to understand the almost normal surfaces for some triangulation.

The sum of two solutions to a set of linear equations is also a solution, so the correspondence between normal/almost normal surfaces and solutions to the equations mentioned above suggests that there should be some way to “add” surfaces. It turns out adding the solutions corresponds to a construction called a Haken sum, in which the loops of intersection between two embedded surfaces are “resolved” to form a single (possibly disconnected) embedded surface. Haken showed that there is a finite collection of solutions (called fundamental solutions) that generate all solutions to the equations. Thus any normal/almost normal surface is a Haken sum of elements from a finite collection of normal/almost normal surfaces.

The Euler characteristic of a Haken sum is the sum of the Euler characteristics of the two surfaces. Thus Rubinstein suggested the following way to look for an algorithm to calculate the Heegaard genus of a 3-manifold: For each Euler characteristic starting from zero, look at all the Haken sums of the fundamental solutions that sum to that number. If one of them is a Heegaard splitting then you’re done. Otherwise, move on to the next Euler characteristic. The problem is that if one of the fundamental solutions has Euler characteristic zero then taking a Haken sum with it does not change the Euler characteristic. There are then an infinite number of solutions one needs to check.

In the paper mentioned above, Lackenby offers the following solution to this problem: He shows that every hyperbolic 3-manifold with toroidal boundary admits a “partially flat, angled ideal triangulation”. This is an ideal triangulation (the interior of the 3-manifold is identified with a simplicial complex minus its vertices) with an angle assigned to each edge of each tetrahedron, with certain conditions (in particular, the angles around each edge must sum to 2\pi). A normal/almost normal surface in this triangulation is cut into polygons whose edges inherit angles from the edges of the tetrahedra. Lackenby shows that the conditions on the angles of the tetrahedra imply that the only normal torus is boundary parallel. This eliminates the problem of having infinitely many normal surfaces of the same Euler characteristic, leading to an algorithm to calculate the Heegaard genus of the 3-manifold.

[2] W. Haken, Theorie der Normalflachen: Ein Isotopickriterium fur der Kreisknoten, Acta Math. 105 (1961) 245–375.
[3] Stocking, Michelle, Almost normal surfaces in $3$-manifolds., Trans. Amer. Math. Soc. 352 (2000), no. 1, 171–207.
[4] J.H. Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds, in ‘Geometric Topology (Athens, GA, 1993)’, Volume 2 of AMS/IP Stud. Adv. Math., 1–20, Amer. Math. Soc., Providence, RI, 1997

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