Knots and Links The Linking Number

We will use this basis for computation. In Chapter 7, we will modify the persistence algorithm to compute canonical cycles and their spanning surfaces. In Chapter 10, we look at data structures and algorithms for computing the linking number of a filtration. Part Two Algorithms 7 The Persistence Algorithms In this chapter, we look at algorithms for computing persistence. We begin by reviewing an algorithm for computing Betti numbers by Delfinado and Edels- brunner 1995 in Section 7.1. This algorithm works over subspaces of S 3 , which do not have torsion. We utilize this algorithm for marking simplices as positive or negative recall Definition 6.3. We also show how the algorithm may be used to speed up the computation of persistence. In Section 7.2, we de- velop the persistence algorithm over Z 2 coefficients for subcomplexes of any triangulation of S 3 . To compute persistence over arbitrary fields, we need the alternate point of view described in Section 6.1.5. Using this view, we extend and generalize the persistence algorithm to arbitrary dimensions and ground fields in Section 7.3. We do so by deriving the algorithm from the classic reduction scheme, illus- trating that the algorithm derives its simple structure from the properties of the underlying algebraic structures. While no simple description exists over non- fields, we may still be interested in computing a single homology group over an arbitrary PID. We give an algorithm in Section 7.4 for this purpose.

7.1 Marking Algorithm

In the first two sections of this chapter, we assume that the input spaces are three-dimensional and torsion-free, as discussed in section 4.2.3. Consequently, we use Z 2 coefficients for computation. Recall from Section 4.3 that using these coefficients greatly simplifies homology: The homology groups are vec- tor spaces, a k-chain is simply the list of simplices with coefficients 1, each simplex is its own inverse, and the group operation is symmetric difference, as shown in Figure 7.1. The only nonzero Betti numbers to be computed are β , β 1 , and β 2 . We also need a filtration ordering of the simplices Definition 2.44. We use 125