Critical Points Stable and Unstable Manifolds
a Stable
b Unstable minimum
saddle maximum
Fig. 5.4. The stable a and unstable b 1-manifolds, with dotted iso-lines h
−1
c, for constants c. In the diagrams, all the saddle points have height between all minima and
maxima. Regions of the 2-cells of maxima and minima are shown, including the critical point, and bounded by the dotted iso-lines. The underlying manifold is S
2
, and the outer 2-cell in b corresponds to the minimum at negative infinity.
minimum saddle
maximum
Fig. 5.5. The Morse-Smale complex of Figure 5.4.
a A single cell on a gray-scale image b Graph of the cell
Fig. 5.6. The Morse-Smale complex of the graph of sin x+siny is a tiling into copies
of the cell shown in a, along with its reflections and rotations. Each cell has simple geometry b.
half of a stable or unstable 1-manifold of a saddle, and each region is a compo- nent of the intersection of a stable 2-manifold of a maximum and an unstable
2-manifold of a minimum.
Example 5.3 sin x + siny Figure 5.6 shows a single cell of the Morse-
Smale complex for the graph of h x, y = sinx + siny. The cell is super-
imposed on a gray-scale image, mapping h x, y to an intensity value for pixel
x, y. The figure shows that each cell has simple geometry: The gradient flows from the maximum to the minimum, after being attracted by the sad-
dles on each side. We saw the Morse-Smale complex for this function on a triangulated domain in Figure 1.9.
6
New Results
This chapter concludes the first part of this book by introducing the nonalgo- rithmic aspects of some of the recent results in computational topology. In
Chapter 1, we established the primary goal of this book: the computational exploration of topological spaces. Having laid the mathematical foundation
required for this study in the previous four chapters, we now take steps toward this goal through
• persistence; • hierarchical Morse-Smale complexes;
• and the linking number for simplicial complexes. The three sections of this chapter elaborate on these topics. In Section 6.1,
we introduce a new measure of importance for topological attributes called persistence
. Persistence is simple, immediate, and natural. Perhaps precisely because of its naturalness, this concept is powerful and applicable in numer-
ous areas, as we shall see in Chapter 13. Primarily, persistence enables us to simplify spaces topologically. The meaning of this simplification, how-
ever, changes according to context. For example, topological simplification of Morse-Smale complexes corresponds to geometric smoothing of the associated
function. To apply persistence to sampled density functions, we extend Morse- Smale complexes to piece-wise linear PL manifolds in Section 6.2. This
extension will allow us to construct hierarchical PL Morse-Smale complexes, providing us with an intelligent method for noise reduction in sampled data.
Finally, in Section 6.3, we extend the linking number, a topological invariant detecting entanglings, to simplicial complexes. Naturally, we care about the
computational aspects of these ideas and their applications. We dedicate Parts Two and Three of this book to examining these concerns.
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