In R The Persistence Module

Fig. 6.5. The inequalities p ≥ 0, l ≥ i, and l + p j define a triangular region in the index-persistence plane. This region defines when the cycle is a basis element for the homology vector space. We give an alternate characterization of this theorem in Chapter 7 while devel- oping the persistence algorithm. By this lemma, computing persistent homol- ogy over a field is equivalent to finding the corresponding set of P-intervals.

6.2 Hierarchical Morse-Smale Complexes

We would like to use persistence to simplify the iso-lines of a 2-manifold and an associated function. But persistence requires a suitably defined filtration. In Chapter 2, we looked at filtrations generated by manifold sweeps. In this section, we will see that the generated filtrations are appropriate for comput- ing persistence and eliminating critical points combinatorially. To modify the function, however, we need control over the geometry. The Morse-Smale com- plex, defined in Chapter 5, provides us with the geometric description that we need. In practice, our function is sampled. This sampling introduces noise into our data and provides the motivation for utilizing persistence for noise-feature differentiation. No matter how dense the sampling, however, our theoretical notions, based on smooth structures, are no longer valid. Triangulating the 2-manifold, we get a piece-wise linear PL function. The gradient of a PL function is not continuous and does not generate the pair-wise disjoint integral lines that are needed to define stable and unstable manifolds. To extend smooth notions to PL manifolds, we use differential structures to guide our computa- tions. We call this method the simulation of differentiability or SoD paradigm. Using SoD, we first guarantee that the computed complexes have the same structural form as those in the smooth case. We then achieve numerical accu- racy by means of transformations that maintain this structural integrity. The separation of combinatorial and numerical aspects of computation is similar to many algorithms in computational geometry de Berg et al., 1997. It is also the hallmark of the SoD paradigm. We show in this section how to extend the ideas from the last chapter to PL manifolds. We will first motivate and define the quasi Morse-Smale complex in Section 6.2.1. A quasi Morse-Smale complex has the same combinatorial structure as the Morse-Smale complex. In Section 6.2.2, I discuss and resolve the artifacts encountered in the PL domain. We then justify the filtrations de- fined in Chapter 2 and relate them to the Morse-Smale complex. We end this section by applying persistence to PL Morse-Smale complexes to get a hierar- chy of progressively coarser Morse-Smale complexes.

6.2.1 Quasi Morse-Smale Complex

We begin by examining the structure of a Morse-Smale complex for a smooth, compact, connected 2-manifold. For brevity, we will call the Morse-Smale complex the MS complex. The following theorem establishes a fact implied by the examples in Chapter 5. Theorem 6.3 quadrangle Each region of the MS complex is a quadrangle with vertices of index 0, 1, 2, 1, in this order around the region. The boundary is possibly glued to itself along vertices and arcs. Proof The vertices on the boundary of any region alternate between saddles and other critical points, which, in turn, alternate between maxima and min- ima. The shortest possible cyclic sequence of vertices around a boundary is therefore 0, 1, 2, 1, a quadrangle. The argument below shows that longer se- quences force a critical point in the interior of the region, a contradiction. Take a region whose boundary cycle has length 4k for k ≥ 2 and glue two copies of the region together along their boundary to form a sphere. Glue each critical point to its copy, so saddles become regular points. Maxima and minima remain as before. The Euler characteristic of the sphere is 2, and so is the alternating sum of critical points, ∑ a −1 i a . However, the number of minima and maxima together is 2k 2, which implies that there is at least one saddle inside the region. Intuitively, a quasi Morse-Smale complex QMS complex, for short is a com- plex with the structural form of a MS complex, as described by Theorem 6.3. The QMS complex is combinatorially a quadrangulation, with vertices at the critical points of h and with edges that strictly ascend or descend as measured by h. But it differs in that its edges may not necessarily be the edges of maxi- mal ascent or descent. Definition 6.9 splitable A subset of the vertices in a complex Q is indepen- dent if no two are connected by an arc. The complex Q is splitable if we can partition the vertices into three sets U ,V,W and the arcs into two sets A, B, so that a U ∪ W and V are both independent; b arcs in A have endpoints in U ∪ V ; and arcs in B have endpoints in V ∪ W , and c each vertex v ∈ V belongs to four arcs, which in a cyclic order around v alternate between A and B. We may then split Q Q splits into two complexes defined by U , A and W, B. Not surprisingly, the MS complex is a splitable quadrangulation. Theorem 6.4 The Morse-Smale complex splits. Proof Following Definition 6.9: a U, V , and W are maxima, saddles, and minima; b set A contains arcs connecting maxima to saddles and set B con- tains arcs connecting minima to saddles; and c saddles have degree 4 and alternate as required. The MS complex then splits into the complex of stable manifolds and the complex of unstable manifolds. A QMS complex splits like the MS complex but does not have the geomet- ric characteristics of that complex. It is like the triangulation of a point set, which has the same combinatorics as the Delaunay triangulation but fails the geometric in-circle test de Berg et al., 1997. Definition 6.10 quasi Morse-Smale complex A splitable quadrangulation is a splitable complex whose regions are quadrangles. A quasi Morse-Smale complex QMS complex of a 2-manifold M and a function h is a splitable quadrangulation whose vertices are the critical points of h and whose arcs are monotone in h. In Chapter 9, we will describe an algorithm for constructing a QMS com- plex, as well as local transformations that transform the complex into the MS complex. a minimum b regular c saddle d monkey e maximum Fig. 6.6. Classifying vertices by their stars. The light-shaded lower wedges are con- nected by white triangles to the dark-shaded upper wedges The dotted vertices and dashed edges on the boundary do not belong to the open star.

6.2.2 Piece-Wise Linear Artifacts

As in the last chapter, we assume that we have a smooth, compact, connected 2-manifold M without boundary, embedded in R 3 . In this section, moreover, we represent the manifold with a triangulation K. We also assume that function h : M → R is linear on every triangle in K. The function is defined, therefore, by its values at the vertices of K. It will be convenient to assume h u = hv for all vertices u = v in K. We simulate simplicity to justify this assumption computationally Edelsbrunner and Mücke, 1990. In order to extend the con- cept of MS complexes to the piece-wise linear domain, we need to look at the artifacts created by the lack of smoothness in a triangulation. Stars. We have already encountered the analog of a neighborhood of a vertex in Section 2.5: the star of a vertex in Definition 2.54, as shown in Figure 6.6. We also looked at the lower and upper stars of a vertex to define filtrations. We may use these to classify a vertex as regular or critical. Definition 6.11 wedge A wedge is a contiguous section of St u that begins and ends with an edge. Fig. 6.7. A monkey saddle may be unfolded into two simple saddles in three different ways. In Figure 6.6, the lower star either contains the entire star or some number k + 1 of wedges, and the same is true for the upper star. If St u = St u, then k = −1 and u is a maximum. Symmetrically, if St u = St u, then k = −1 and u is a minimum. Otherwise, u is regular if k = 0 and a saddle if k = 1. Unlike the smooth case, monkey saddles and even more complicated configurations are possible in triangulations. Definition 6.12 multiple saddle A vertex u is a k-fold saddle or a saddle with multiplicity k if St u has k + 1 wedges. A 2-fold saddle is often called a monkey saddle . For k ≥ 2, k-fold saddles are also called multiple saddles. We can unfold a k-fold saddle into two saddles of multiplicity 1 ≤ i, j k with i + j = k by the following procedure. We split a wedge of St u through a triangle, if necessary and similarly split a nonadjacent wedge of St u. The new number of lower and upper wedges is 2 k + 1 + 2 = 2i + 1 + 2 j + 1, as required. By repeating the process, we eventually arrive at k simple saddles. The combinatorial process is ambiguous, but it is usually sufficient to pick an arbitrary unfolding from the set of possibilities. There are three minimal unfoldings for a monkey saddle, as shown in Figure 6.7. Merging and forking. The definition of integral lines is inherently dependent on the smoothness of the space. In their place, we construct monotonic curves that never cross in K. Such curves can merge together and fork after a while. Moreover, it is possible for two curves to alternate between merging and fork- ing an arbitrary number of times. To resolve this, when two curves merge, we will pretend that they maintain an infinitesimal separation, running side by side without crossing. Figure 6.8 illustrates the two PL artifacts and the corre- sponding simulated smooth resolution. As always, we will only simulate the smooth resolution combinatorially. a Merge b Smooth flow c Fork d Smooth flow Fig. 6.8. Merging a and forking c PL curves and their corresponding smooth flow pictures b, d. Fig. 6.9. Nontransversality: The unstable 1-manifold of the lower saddle approaches the upper saddle. Nontransversal intersections. Another artifact of PL domains is nontransversal intersections. We illustrate this artifact via the standard ex- ample in Morse theory: the height function over a torus, standing on its side. The lowest and highest points of the inner ring are the only saddles, as shown in Figure 6.9. Both the unstable 1-manifold of the lower saddle and the sta- ble 1-manifold of the upper saddle follow the inner ring and overlap in two open half-circles. The characteristic property of a nontransversal intersection is that the unstable 1-manifold of one saddle approaches another saddle, and vice versa. Generically, such nontransversal intersections do not happen. If they do happen, an arbitrarily small perturbation of the height function suffices to make the two 1-manifolds miss the other saddles and approach a maxi- mum and a minimum without meeting each other. The PL counterpart of a nontransversal intersection is an ascending or descending path that ends at a saddle. Once again, we will simulate the generic case by extending the path beyond the saddle.