Today I want to write about an application of axiomatic/iterated thin position that connects thin position very strongly to the theory of normal/almost normal surfaces. This is the application that actually got me started thinking about thin position, and which I’m writing up in paper-form as I write these posts. This will probably be my last post about thin position for a while, so before beginning my next post on axiomatic thin position, here’s a quick synopsis of what I’ve covered so far:
1. (2/6/09) I defined a cell complex called the complex of surfaces whose vertices were isotopy classes of surfaces in a given 3-manifold. The disk complex for each surface can be found in the link of the corresponding surface.
2. (2/10/09) I introduced three axioms that encoded properties of this cell complex and began defining thin position in terms of the complex.
3. (2/17/09) I definined orientations on the edges of the complex of surfaces that, along with a couple of new axioms, allowed me to define Scharlemann-Thompson thin position in terms of the complex of surfaces.
4. (2/24/09) I defined the complex of paths (in the complex of surfaces), which satisfies the axioms from part 2, allowing one to define thin position for paths and thus “iterating” the thin position arguments.
5. (3/3/09) I introduced a final axiom about the complex of surfaces that allows one to characterize thin paths in the complex of paths. I then outlined how this leads to Bachman’s proof of the Gordon conjecture.
The application that I will write about today is a generalization of the Hayashi-Shimokawa thin position that I mentioned in post number 4: Let M be a 3-manifold and G a graph, a link or a disjoint union of graphs and links (why not?) embedded in M. Vertices of the complex of surfaces S(M, G) for the pair (M, G) will be strongly separating surfaces in M that are transverse to G, modulo isotopies of the surface that keep it transverse to G. Edges will correspond to compressing a surface in the complement of G (along a loop that is essential in the surface minus G, but not necessarily in the surface itself), bridge compressing a surface, removing a trivial sphere component from the surface, or pushing the surface across a vertex in G. The cells of S(M, G) are defined by sets of edges corresponding to moves that commute with each other, as in the original complex of surfaces. (I’ll leave the details to the reader.)
The point of this construction is not to study the properties of G, but to turn the problem of understanding surfaces in M into a problem of understanding surfaces in the complement of G. So a careful choice of G should allow one to prove things about surfaces in M. Here are three examples of how one might choose G, along with conjectures for the sorts of theorems that should result:
If M is a Seifert fibered space then let G be the union of its critical fibers. The complement in M of G is then a surface cross a circle. It should be possible to recover the classification of Heegaard splittings of Seifert fibered spaces  from this picture. Also, thanks to iterated thin position, I bet one can even find a new way of distinguishing between different isotopy classes of Heegaard splittings in Seifert fibered spaces. (This would be a purely combinatorial/topological proof, as opposed to Lustig and Moriahs very algebraic methods .)
Given a Heegaard splitting for a 3-manifold M, one could let G be the union of the spines of the handlebodies in the Heegaard splitting. The complement of G would then be a surface cross an interval, which is again a very simple space. Given a strongly irreducible or critical surface with respect to this G that intersects both spines, one should be able to bound the Hempel distance of the original Heegaard splitting. This should lead to a new proof of Scharlemann-Tomova’s genus vs. distance result  and to my result bounding the distance of flippable Heegaard splittings , without using the Rubinstein-Scharlemann graphic.
Finally, given a triangulation of M, one can let G be the 1-skeleton of the triangulation. In this situation, I know what happens because this is what I’ve been writing up in paper form. It turns out that with this setup, the index-zero vertices correspond to normal surfaces, the index-one vertices correspond to almost normal surfaces, and the index-two vertices correspond to a class of surfaces called index-2 normal surfaces. Such surfaces can actually be calculated algorithmically from a (reasonably nice) triangulation. Moreover, there is an algorithm to determine which of these surfaces (as vertices in the complex of surfaces) are connected by descending, oriented paths. This means that there is an algorithm that can construct enough of the complex of surfaces for (M, G) to work out all the paths, i.e. a list of Heegaard splittings based on the triangulation.
Thanks to iterated thin position, isotopies of Heegaard splittings show up in this picture as well, defined by the index-2 normal surfaces. So, given a reasonably nice triangulation of M, such as one of Marc Lackenby’s partially flat angled ideal triangulation , one can make a list of Heegaard splittings such that each isotopy class with genus below some constant appears exactly once. The details of this are in the works and I hope to have something on the front by the summer.
 Yoav Moriah and Jennifer Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998), no. 5, 1089-1112.
 Martin Lustig and Yoav Moriah, Nielsen equivalence in Fuchsian groups and Seifert fibered spaces, Topology 30 (1991), no. 2, 191-204.