Someone recently pointed out to me a paper by A. J. Pajitnov  proving a very interesting connection between circular Morse functions and (linear) Morse functions on knot complements. (A similar result is probably true in general three-manifolds as well.) Recall that a (linear) Morse function is a smooth function from a manifold to the line in which there are a finite number of critical points (where the gradient of the function is zero), and each critical point has one of a number of possible forms. For a two-dimensional manifold the possible forms are the familiar local minimum, saddle or local maximum. This post is about three-dimensional Morse functions, in which case the possible forms are slight generalizations of local minima, maxima and saddles. A circular Morse function is a function with the same conditions on critical points, but whose range is the circle rather than the line. For a three-dimensional manifold, the minimal number of critical points in a linear Morse function is twice the Heegaard genus plus two, and for knot complements it’s twice the tunnel number plus two. (In particular, one can construct a Heegaard splitting or unknotting tunnel system directly from a Morse function, but that’s for another post.) The minimal number of critical points in a circular Morse function is called the Morse-Novikov number, and is equal to the minimal number of handles in a circular thin position for the manifold (usually a knot complement). Pajitnov has a very clever argument to show that the (circular) Morse-Novikov number of a knot complement is bounded above by twice its (linear) tunnel number. Below, I want to outline a slightly different formulation of this proof in terms of double sweep-outs, though I should stress that the underlying idea is the same.
Given a genus Heegaard splitting for a three-manifold , one can construct a Morse function with critical points, namely one index-zero critical point (a local minimum), index-one critical points, index-two critical points and one index-three critical point (a local maximum). If the first cohomology group of is infinite (and thanks to Poincare duality, this is equivalent to having infinite first and/or second homology groups), there is a map that realizes a generator of this group. For a knot complement, the first cohomology group is , so there is a canonical generator which is dual to any Seifert surface. The question is, how do we turn the critical points in the linear Morse function into critical points of a circular Morse function ?
Pajitnov’s proof shows that you can adjust the gradient field defined by to the gradient field of a circular Morse function without introducing any new critical points. But we can do essentially the same thing with double sweep-outs. First let be any circular Morse function for and consider the map . In other words, we define . By employing tricks from singularity theory (such as in ), we can isotope and so that this map is stable, a generalization of the Morse conditions to higher dimensional range spaces. If is stable then the set of critical points (where the discriminant of the function is not onto) is a one-dimensional smooth submanifold of called the discriminant set (or sometimes the Jacobi set). The image in of the discriminant set is a graph called that graphic. (There are no vertices in the discriminant set, but its map to will generally be two-to-one at finitely many points, which form vertices in the graphic.)
It turns out that you can read a lot of information about and from the graphic. For example, as I pointed out in , the critical points of and correspond to vertical tangencies and horizontal tangencies, respectively, of edges of this graph. (This was for two linear Morse functions, but the same is true in this setting.) When I write horizontal and vertical, I’m thinking of the circle factor of as being vertical and the factor as horizontal. Also, note that the edges of this graph will not be straight lines, but curves with isolated tangents.
In this picture, each of and is a composition of with a projection of onto or . In particular, is a composition with a vertical projection and is a composition with a horizontal projection. The critical points of the composition correspond to the points where edges of the graphic are tangent to the direction of the projection. If we choose a different projection, we’ll get a different function and we can use this fact to characterize the critical points of the new function.
In particular, define a projection from to by where , , is a very large number and is taken modulo 1 (or rather modulo whatever the circumference of is). I like to think of this as applying a vertical twist to the circles in then projecting horizontally. (This is roughly what happens to a slinky if you hold the bottom steady and twist the top, except our slinky is on its side.) As becomes larger, the slope in that corresponds to being tangent to the projection becomes closer to vertical. Because the graphic is smooth, there will be some slope -close to vertical such that every point with that slope is adjacent to a unique vertical tangency, i.e. a critical point of . So if we pick the corresponding for our projection then the composition of with will be a circular Morse function with the same number of critical points as .
Note that we can’t go the other way, i.e. find a linear Morse function whose number of critical points is bounded by the Morse-Novikov number, because there is no way to “spin” around the factor. For example there are fibered knots (with Morse-Novikov number zero) with arbitrarily large tunnel number.
 Kobayashi, Tsuyoshi; Saeki, Osamu The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map. Pacific J. Math. 195 (2000), no. 1, 101–156.