A knot in **R**^3 is in bridge position if the horizontal plane (y = 0, say) cuts it into two sets of arcs, each of which can be isotoped into the plane (though both can’t be isotoped into the plane at the same time). This is roughly equivalent to the condition that in the restriction of the height function f(x,y,z) = y to the knot, every maximum is above every minimum. This definition suggests a connection between bridge positions and Heegaard splittings since a Heegaard splitting can be defined as a level set of a Morse function in a 3-manifold in which every index *i* critical point is below every index *(i + 1)* critical point. (The Heegaard surface sits between the index 2 and index 3 critical points.) In this definition, we don’t care if the Morse function is induced by an embedding of the 3-manifold into a higher dimensional space. But what if we did?

Scharlemann [1], [2] has used an idea along these lines in his work on the Schoenflies conjecture (that every 3-sphere smoothly embedded in **R**^4 bounds a smooth ball), showing, in particular, how to interpret the embedding into **R**^4 as a picture in **R**^3. Scott Taylor [3], [4] has studied the picture in **R**^3 in more detail. Of course, Scharlemann focussed entirely on embeddings of the 3-sphere. It might be interesting to apply these sorts of methods to Heegaard splittings of other 3-manifolds. (Note: The link from [1] was originally to the wrong paper, but it’s fixed now.)

For example, given a 3-manifold that can be embedded in **R**^4, can it be embedded so that the restriction of the height function is a Morse function inducing a minimal genus Heegaard splitting? Can every embedding be isotoped so the height function induces a minimal genus Heegaard splitting? For 3-manifolds that can’t be embedded in **R**^4, one can ask the question for the smallest **R**^n that it can be embedded in.

My guess is that the answer is no, but I don’t know how one might prove it. (I’d start with Scharlemann’s methods for the 3-sphere in **R**^4.) Note that if a 3-manifold can be embedded in **R**^n then it can be embedded in **R**^(n+1) in a way that can be isotoped to induce any Heegaard splitting. It might also be interesting to try to characterize which gluing maps produce Heegaard splittings that come from embeddings in **R**^4, **R**^5, etc.

[2] M. Scharlemann, Smooth spheres in R4 with four critical points are standard, Invent. Math. 79 (1985) 125–141.

You mean that it bounds a smooth ball, right?

Comment by blueollie — June 15, 2008 @ 5:04 pm |

Thanks. I’ve corrected it above.

Comment by Jesse Johnson — June 16, 2008 @ 8:47 am |