a b
b p
p
a
a b
p
b=p
b
Fig. 9.9. The two cases in the algorithm. The iso-line is dotted, the annuli are shaded, the arcs bounding the annuli are bold dashed, and the new paths emanating from a are
bold solid.
to be changed first, which we do by recursive application of the algorithm, as for the lower new path in Figure 9.9a.
It is also possible that p is more than one position removed from b, as for the upper new path in Figure 9.9b. In this case we perform several slides for
a , the first connecting a to the first maximum after b in the direction of p. Each
such slide may require recursive slides to clear the way, as before. Finally, it is possible that the new path from a to p winds around the arc boundary of
the annulus several times, as does the lower new path in Figure 9.9b. The algorithm is the same as before.
The winding case shows that the number of slides cannot be bounded from above in terms of the number of critical points. Instead, consider the crossings
between arcs of the initial QMS and the final MS complexes, and note that the number of slides is at most some constant times the number of such crossings.
10
The Linking Number Algorithm
In Chapter 6, we discussed a topological invariant called the linking number and extended this invariant to simplicial complexes. In this chapter, we provide
data structures and algorithms for computing the linking numbers of a filtra- tion, using the canonical cycles and manifolds generated by the persistence
algorithm. After motivating this computation, we describe the data structures and algorithms. We end this chapter by discussing an alternate definition of the
linking number that may be helpful in understanding the topology of molecular structures.
10.1 Motivation
In the 1980s, it was shown that DNA, the molecular structure of the genetic code of all living organisms, can become knotted during replication Adams,
1994. This finding initiated interest in knot theory among biologists and chemists for the detection, synthesis, and analysis of knotted molecules Fla-
pan, 2000. The impetus for this research is that molecules with nontrivial topological attributes often display exotic chemistry. Such attributes have been
observed in currently known proteins. Taylor recently discovered a figure-of- eight knot in the structure of a plant protein by examining 3,440 proteins using
a computer program Taylor, 2000. Moreover, chemical self-assembly units are being used to create catenanes, chains of interlocking molecular rings,
and rotaxanes, cyclic molecules threaded by linear molecules. Researchers are building nano-scale chemical switches and logic gates with these struc-
tures Bissell et al., 1994; Collier et al., 1999. Eventually, chemical computer memory systems could be built from these building blocks.
171
10.1.1 Prior work
Catenanes and rotaxanes are examples of nontrivial structural tanglings. The focus of this chapter is on computing the linking number, the link invariant de-
fined in Section 6.3. Haken 1961 showed that important knotting and linking problems are decidable in his seminal work on normal surfaces. His approach,
as reformulated by Jaco and Tollefson 1995, forms the basis of many cur- rent knot detection algorithms. Hass et al. 1999 showed that these algorithms
take exponential time in the number of crossings in a knot diagram. They also placed both the
UNKNOTTING PROBLEM
and the
SPLITTING PROBLEM
in NP, the latter problem being the focus of this chapter. Generally, other approaches
to knot problems have unknown complexity bounds and are assumed to take at least exponential time. As such, the state of the art in knot detection only
allows for very small data sets.
10.1.2 Approach
The approach in this chapter is to model molecules by filtrations of α-complexes, and detect potential tanglings by computing the linking num-
bers of the filtration. The linking numbers constitute a signature function for the filtration. This combinatorial approach makes the same fundamental as-
sumption as in Chapter 2 that
α-complex filtrations capture the topology of a molecular structure. Given a filtration, we will use the spanning surface defi-
nition of the linking number for its computation. Consequently, we need data structures for the efficient enumeration of co-existing pairs of cycles in differ-
ent components. We also need an algorithm to compute the linking number of a pair of such cycles.
10.2 Algorithm
In this section, we present data structures and algorithms for computing the linking numbers of the complexes in a filtration. As we only use canonical
1-cycles for this computation, we will refer to them simply as cycles. As- sume we have a filtration K
1
, K
2
, . . . , K
m
as input. As simplices are added, the complex undergoes topological changes that affect the linking number: New
components are created and merged together, and new nonbounding cycles are created and eventually destroyed. A basis cycle z with persistence interval
[i, j may only affect the linking numbers of complexes K
i
, K
i +1
, . . . , K
j
−
1
in the filtration. Consequently, we only need to consider basis cycles z
′
that exist during some subinterval
[u, v ⊆ [i, j in a different component than z’s.
