In the last few posts, I’ve written about an axiomatic approach to thin position, based on an object I call the complex of surfaces. Last time I showed how to reconstruct Scharlemann-Thompson thin position for 3-manifolds using this approach. For further applications, there are a number of closely related complexes that satisfy the axioms that I’ve defined: One could take vertices to be embedded 2-spheres in the 3-sphere, modulo isotopy transverse to a given knot, and have edges correspond to pushing an arc of the knot through one of these spheres. Applying the method described so far would recover Gabai’s original definition of thin position for knots . One could also combine this knot thin position with 3-manifold thin position: Let vertices be arbitrary (strongly separating) surfaces modulo isotopy tranverse to a knot in some 3-manifold, with edges corresponding to compressions in the complement of the knot and to pushing arcs of the knot through the surface. The result would be Hayashi and Shimokawa’s flavor of thin position . With a few more tweaks, one could probably recover Tomova’s thin position for knots (which is more compatible with the curve complex) .
But I’ll leave all that as an exercise for the reader. Today I want to write about a less obvious application of the axiomatic method, which I call iterated thin position. Here’s the idea: Start with a complex S that satisfies all the axioms we’ve defined so far. A splitting path in this complex has a well defined complexity, as described already, given by the complexities of its maximal vertices, and we can order these paths with lexicographic ordering of their complexities. We also have some natural moves on splitting paths: sliding across squares and adding or removing length two loops.
Define P(S) to be the complex whose vertices are equivalence classes of splitting paths in S. We’ll okace edges between paths that are related by a single one of these moves. If there is a representative path in S for a vertex of P(S) such that some number of moves can be done simultaneously (i.e. the squares in S that we want to slide across are disjoint) then we add a cube to P(S) defined by the edges in P(S) corresponding to those moves. We will call this complex the complex of (splitting) paths for S.
So, this gives us a complex with an ordering on its vertices. The reader can check that P(S) satisfies the strictness axiom, the parallel axiom and the minima axiom that I defined a couple of weeks ago. Unlike the complex of surfaces itself, there doesn’t seem to be any way of assigning orientations to the edges of P(S), but that’s ok. Let’s see what we can do with the three axioms we do have.
First, let’s consider what it means for a vertex in P(S) to have index zero, i.e. no edges descending from it. Recall that in S, a there are two thinning moves (i.e. moves that reduce the complexity of the path): Removing a length two loop and pushing a maximum of the path down by a vertical slide. If a vertex in P(S) has no edges descending from it then for any path in the equivalence class it represents, there are no thinning moves. In other words, a path represented by such a vertex is slender. As we saw last time, the minima of such a path are index zero in S and the maxima are strongly irreducible.
Since we’re not worried about oriented paths in P(S), we should next look at index one vertices (rather than strongly irreducible vertices). Recall that a vertex in a complex satisfying the first three axioms has index one if its descending link is disconnected. In P(S), a vertex in the descending link corresponds to a thinning move on a path in S. A thinning move always happens at a maximum in the path and if there are two thinning moves at two different maxima then these can always be done simultaneously. Thus if a path has two maxima where it can be thinned, the two thinning moves define vertices in the same component of the descding link. Any other thinning move for that path is connected to one of those two thinning moves by an edge so the descending link must be connected. Thus if a vertex of P(S) has index one then the corresponding path in S has exactly one maximum that is not strongly irreducible.
We would now like to characterize the one maximum that is not strongly irreducible, and perhaps the minima of a path representing an index-one vertex in P(S). To do that we need one more axiom on S, which I will write about next week. For now, I want to point out that what we’re recovering from P(S) is precisely Dave Bachman’s theory of sequences of generalized Heegaard splittings. I wrote some entries about these a few months ago and got sufficiently muddled that I don’t think they’re worth reading (let along including a link). But with this axiomatic approach, I’m feeling more optomistic. Next time I should be able to outline Dave’s proof of the Gordon conjecture using this iterated thin position machinery.
 Gabai, David Foliations and the topology of $3$-manifolds. II. J. Differential Geom. 26 (1987),no. 3, 461–478. 57N10 (57R30)
 C. Hayashi and K. Shimokawa, “Heegaard splittings of the pair of the solid torus and the core loop”, Rev. Mat. Complut. 14:2 (2001), 479–501. MR1871309 (2003a:57038)