Hierarchy Hierarchical Morse-Smale Complexes

−1 +1 +1 −1 a A link diagram for the Whitehead link +1 −1 b Crossing label convention Fig. 6.16. The Whitehead link a is labeled according to the convention b that the crossing label is +1 if the rotation of the overpass by 90 degrees counter-clockwise aligns its direction with the underpass, and −1 otherwise.

6.3.2 The Linking Number

As before, we may use invariants as tools for detecting whether a link is sep- arable. Seifert first defined an integer link invariant, the linking number, in 1935 to detect link separability Seifert, 1935. There are several equivalent definitions for the linking number. I give the most accessible definition below for intuition. Given a link diagram for a link l, we first choose orientations for each knot in l. We then assign integer labels to each crossing between any pair of knots k , k ′ , following the convention in Figure 6.16b. Let λk, k ′ of the pair of knots to be one-half the sum of these labels. A standard argument using Reidermeister moves shows that λ is an invariant for equivalent pairs of knots up to sign. Definition 6.18 linking number The linking number λl of a link l is λl = ∑ k =k ′ ∈ l |λk, k ′ |, 6.5 where λk, k ′ is one-half the sum of labels on oriented knots k, k ′ according to the convention in Figure 6.16b. Note that λl is independent of knot orientations. Also, the linking number has the characteristic of invariants that it does not completely recognize link- ing. The Whitehead link in Figure 6.16a, for example, has linking number zero but is not separable. If the linking number is nonzero, however, we know that the link is not the unlink. I will use an alternate definition for developing algorithms for computing the linking number in Chapter 10. This definition is based on surfaces whose boundaries are the knots in the link. Fig. 6.17. The Hopf link and Seifert surfaces of its two unknots are shown on the left. Clearly, λ = 1. The spanning surface for the cycle on the right is a Möbius strip and therefore nonorientable. Definition 6.19 spanning, Seifert A spanning surface for a knot k is an em- bedded surface with boundary k. An orientable spanning surface is a Seifert surface . Figure 6.17 shows examples of spanning surfaces for the Hopf link and Möbius strip. Since a Seifert surface is orientable, we may label its two sides as positive and negative. Given a pair of oriented knots k , k ′ and a Seifert surface s for k, we label s by using the orientation of k. We then adjust k ′ via a homotopy h until it meets s in a finite number of points. Following along k ′ according to its orientation, we add +1 whenever k ′ passes from the negative to the positive side and −1 whenever k ′ passes from the positive to the negative side. The following theorem asserts that this sum is independent of our the choice of h and s, and it is, in fact, the linking number. Theorem 6.6 Seifert surface λk, k ′ is the sum of the signed intersections between k ′ and any Seifert surface for k. The proof is by the standard Seifert surface construction. If the spanning sur- face is nonorientable, we can still count how many times we pass through the surface, giving us the following weaker result. Theorem 6.7 spanning surface λk, k ′ mod 2 is the parity of the num- ber of times k ′ passes through any spanning surface for k.

6.3.3 Graphs

In order to compute the linking number of a simplicial complex, we need to first define what we mean by a knot in a complex. Not surprisingly, we decide to use the homology cycles of a simplicial complex, as defined in Chapter 4. a K 800 b Graph of homology cycles in K 800 Fig. 6.18. The homology cycles of the 800th complex K 800 of a filtration for data set 1grm a form a graph b. The darker negative edges form a spanning forest that defines a canonical basis for the cycles. These cycles form a graph within the simplicial complex, as shown in Fig- ure 6.18. We need to extend the linking number to graphs, in order to use the theorems in the last section in computing linking numbers for simplicial complexes. Let G = V, E, E ⊆ V 2 be a simple undirected graph in R 3 with c com- ponents G 1 , . . . , G c . A graph may be viewed as a vector space of cycles. For example, the graph in Figure 6.18b has rank 35. Let z 1 , . . . , z m be a fixed basis for the cycles in G, where m = |E| − |V |+c is the rank of G. We then define the linking number between two components of G to be λG i , G j = ∑ |λz p , z q | for all cycles z p , z q in G i , G j , respectively. The linking number of G is then defined by summing the total interactions between pairs of components. Definition 6.20 linking number of graphs The linking number λG of a graph G is λG = ∑ i = j λG i , G j , where λG i , G j = ∑ |λz p , z q | for ball basis cycles z p , z q in different compo- nents G i , G j , respectively. The linking number is computed only between pairs of components following Seifert’s original definition. Linked cycles within the same component may be unlinked by a homotopy Prasolov, 1995. G 1 G 2 a Graph G = G 1 ∪ G 2 G G 1 2 b λG = 1 G G 1 2 c λG = 2 Fig. 6.19. We get different λG for graph G a depending on our choice of basis for G 2 : two small cycles b or one large and one small cycle c. σ Fig. 6.20. Solid negative edges combine to form a spanning tree. The dashed positive edge σ creates a canonical cycle. Figure 6.19 shows that the linking number for graphs is dependent on the chosen basis. While it may seem that we want λG = 1 in the figure, there is no clear answer in general. We need a canonical basis for defining a canonical linking number. The definition of the canonical basis is similar to the one used for the fundamental group of a graph Hatcher, 2001. Recall that persistence marks simplices as positive or negative, depending on whether they create or destroy cycles. Each negative edge connects two components. Therefore, the set of all negative edges gives us a spanning forest of the complex, as shown in Figures 6.20 and Figure 6.18b. Every time a positive edge σ is added to the complex, it creates a new cycle. We choose the unique cycle that contains σ and no other positive edge as a new basis cycle. Definition 6.21 canonical The unique cycle that contains a single positive edge is a canonical cycle. The set of all canonical cycles is the canonical basis .