Low Dimensional Topology

January 15, 2008

Width Complexes for Knots and 3-Manifolds

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 1:11 pm

In the first talk at last week’s special session on Heegaard splittings, etc. at the joint meetings, Jennifer Schultens discussed something she calls the width complex for a knot or link. The idea is that one should consider all “positions” of the link with respect to a sweep-out for the ambient manifold. If the restriction of the sweep-out to the knot is Morse then there is a family of level surface that separate the critical points into blocks of maxima and minima. These give a thick-thin decomposition of the link. (I don’t know if there’s a standard word for this family of surfaces, so I’ll just call them a thick-thin decomposition until someone corrects me.) One can define a complexity for such a position (called the width) by considering the number of intersections of the knots with the surfaces. A thin position for the link is a thick-thin decomposition of minimal width and a bridge position is one with a single surface. (See thin position for a definition of width.) One normally would consider the width complex defined by a genus zero sweep-out of the 3-sphere, but there’s no reason to restrict things so much.

There is a simple collection of (two) moves that can be used to go from any thick-thin decomposition to any other (I’ll let the reader work them out.) so one can define a complex (or at least a graph) of thick-thin decompositions. Schultens uses a slightly different construction than the one above to define thick-thin decompositions (though the resulting complex is the same) and uses Reidemeister moves to prove that its connected. Looking at the path of Morse functions on the circle induced by an isotopy of the knot suggests a second proof that it’s connected. The width complex has infinite diameter because the bridge number of a thick-thin decomposition changes by at most one along each edge and there are positions with arbitrarily high bridge number (which implies arbitrarily high width). Schultens asks if one can get arbitrarily far apart without the width going to infinity. In other words, is there a family thick-thin decompositions of a given knot that have bounded width but unbounded diameter in the width complex?

Here’s another question one can ask about this complex, though this may be going out on a limb a little. A path in the width graph/complex corresponds to an isotopy of the knot, i.e. a path in Hatcher’s space of knots. If one wanted to add higher dimensional cells to the width complex, one could look at isotopies of paths in the space of knots and look at how the induced paths in the width complex change when the isotopy passes through non-generic paths. (This is roughly the idea in Hatcher and Thurston’s construction of a presentation for the mapping class group of a surface [1].) This should suggest some 2-cells and after gluing in these 2-cells, the resulting complex should have the same fundamental group as the space of knots. One can similarly define higher dimensional cell and the resulting complex should have the same higher homotopy groups as the space of knots. What I want to know is: is the width complex with these higher dimensional cell homotopy equivalent to the space of knots?

One can make analogous definitions for generalized Heegaard splittings of 3-manifolds and construct a similar complex (though I don’t think it’s obvious what the 2-cells should be). David Bachman has been implicitly using such an idea in his definition of sequences of generalized Heegaard splittings (or SOGs). A SOG is a path in the complex of generalized Heegaard splittings. Bachman has suggested that in order to compare two Heegaard splittings, one should find the path that minimizes the complexity of the generalized Heegaard splittings through which it passes. Then the local maxima along the path should have certain nice properties (He calls these critical splittings.) He first suggested this approach as a way to prove the Gordon conjecture. (This conjecture states that a connect sum of two irreducible Heegaard splittings should also be irreducible. ) Bachman’s preprint on this conjecture has gone through a number of versions, suggesting that his definition of critical may still need some work. His examples of Heegaard splittings that require many stabilizations to flip (mentioned in a previous post) rely on similar techniques, and again I think the definition needs some ironing. However, the general idea of choosing the most efficient path in the width complex seems to have potential and if Bachman or someone else finds a good definition of critical and a clear way to exploit it, this could prove to be a very useful technique.

[1] A. Hatcher and W. Thurston, A presentation for the mapping class group of a
closed orientable surface., Topology 19 (1980), no. 3, 221{237.

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