ordered by persistence forms an arc in the complex obtained by canceling the first i
− 1 pairs. Proof
Assume without loss of generality that the i-th pair consists of a negative saddle a
= u
j +1
and a positive minimum z. Consider the component of K
j
that contains z. One of the descending paths originating at a enters this component,
and because it cannot ascend, it eventually ends at some minimum b in the same component. Either b
= z, in which case we are done, or b has already been paired with a saddle c
= a. In the latter case, c has height less than a; it belongs to the same component of K
j
as b and z; and the pair b , c is one of
the first i − 1 pairs of critical points. It follows that when b gets canceled, the
path from a to b gets extended to another minimum d, which again belongs to the same component. Eventually, all minima in the component other than z are
canceled, implying that the initial path from a to b gets extended all the way to z
. The claim follows. We may cancel pairs of critical points combinatorially without the need of
an MS complex, using the simplification algorithms given in Chapter 8. For simplifying terrains, however, we would like to modify the geometry so that
critical points actually disappear. The MS complex provides us with the geo- metric control we need for this modification.
6.3 Linking Number
In the last two sections, we described a measure for topological attributes and showed how it may be applied to simplify a sampled density function. In
this section, we discuss another topological property: linking. Figure 6.15 shows the five linked tetrahedral skeletons we last saw in Chapter 1. Intuitively,
we say an object is linked if components of the object cannot be separated from each other. In this section, we consider the linking number, a topological
invariant that detects linking. As before, we are interested in computing linking in a filtration. To do so, we need to extend the definition of the linking number
to simplicial complexes.
The mathematical background needed for this section is rather brief, so I present it here in the first two sections instead of placing it in a separate chap-
ter. My treatment follows Adams 1994, a highly readable introductory book, as well as Rolfsen 1990, the classic textbook on knots and links. The last
section includes new results. I extend the linking number to graphs and define a canonical basis for the set of homological 1-cycles in a simplicial complex.
Fig. 6.15. The skeletons of 5 regular tetrahedra defined by the 20 vertices of the regular dodecahedron. The tetrahedra are linked pair-wise.
6.3.1 Knots and Links
We begin by examining a few basic definitions of knot theory.
Definition 6.14 knot A knot is an embedding of a circle in three-dimensional Euclidean space, k : S
1
→ R
3
. That is, k does not have self-intersections. As before, we define an equivalence
relation on knots in order to classify their topologies.
Definition 6.15 knot equivalence Two knots are equivalent if there is an ambient isotopy that maps the first to the second.
In other words, we may deform a knot to an equivalent knot by a continuous motion in R
3
that does not cause intersections in the knot at any time.
Definition 6.16 link A link l is a collection of knots with disjoint images.
For example, the union of two circles whose projections onto a plane are dis- joint is a link called the unlink.
Definition 6.17 separable A link is separable splitable if it can be contin- uously deformed via an ambient isotopy so that one or more components can
be separated from the other components by a plane that itself does not intersect any of the components.
The unlink is separable; linked knots are not. We often visualize a link l by a link diagram, a the projection of a link onto a plane, such that the over- and
undercrossings of knots are presented clearly. Figure 6.16a is one commonly used diagram of the Whitehead link. The knots in the figure are also oriented
arbitrarily. For a formal definition of a link diagram, see Hass et al., 1999.
−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.
