Table 12.9. Time in seconds for canonizing 1-cycles and 2-cycles, along with the average cycle length and spanning manifold size, before and after
canonization.
1-cycles 2-cycles
cycle len manifold size
cycle len manifold size
time before
after before
after time
before after
before after
torus 0.01
4.36 11.34
1.55 45.85
0.01 6.85
34.68 2.94
36.56 hopf
0.02 4.06
52.03 1.06
62.57 0.01
5.00 14.37
1.00 10.44
SOD 0.02
4.89 15.86
2.18 41.25
0.01 6.36
18.48 2.28
16.16 1grm
0.02 4.26
14.05 1.38
51.38 0.01
5.24 18.52
1.17 15.87
möbius 0.03
4.14 28.97
1.14 62.10
0.01 5.00
17.42 1.00
24.42 LTA
0.16 4.91
24.94 1.98
111.64 0.05
5.84 24.44
1.63 18.79
1mbn 0.19
4.44 48.68
1.55 115.73
0.05 5.24
19.79 1.16
16.69 FAU
0.13 4.39
25.70 1.40
78.60 0.06
5.59 26.62
1.57 27.88
KFI 0.13
4.67 27.88
1.69 66.79
0.05 6.31
19.91 1.93
17.12 1qb0
0.28 4.50
60.56 1.63
141.52 0.05
5.29 20.24
1.20 16.71
BOG 0.23
5.67 25.68
3.21 112.71
0.07 5.78
22.43 1.51
22.56 1hiv
0.30 4.50
57.21 1.64
124.55 0.06
5.31 20.74
1.22 17.36
1hck 0.57
4.49 68.71
1.63 166.61
0.11 5.29
24.29 1.20
22.72 bearing
1.95 4.78
56.98 2.53
583.15 0.17
5.81 22.87
1.67 26.68
dragon1 1.01
4.21 43.23
1.25 226.23
0.55 5.17
48.40 1.19
66.81 TAO
1.69 4.46
40.94 1.81
160.72 0.35
5.31 22.35
1.21 21.24
Fig. 12.6. A wire-frame visualization of dataset K
, an immersed triangulated Klein bottle with 4000 triangles.
last section, when we do not use the union-find speedup. We use a 2.2 GHz Pentium 4 Dell PC with 1 GB RAM running Red Hat Linux 7.3 for computing
the timings in this section.
12.3.2 Framework and Data
We have implemented a general framework for computing persistence com- plexes from Morse functions defined over manifolds of arbitrary dimension.
Our framework takes a tuple K, f as input and produces a persistence com-
plex CK, f as output. K is a d-dimensional simplicial complex that triangu-
lates an underlying manifold. And f : vert K → R is a discrete function over
the vertices of K that we extend linearly over the remaining simplices of K. The function f acts as the Morse function over the manifold, but it need not be
Morse for our purposes, as we perform symbolic perturbation to eliminate the degeneracies. Frequently, our complex is augmented with a map
ϕ : K → R
d
that immerses or embeds the manifold in Euclidean space. Our algorithm does not require
ϕ for computation, but ϕ is often provided as a discrete map over the vertices of K and is extended linearly as before. For example, Figure 12.6
displays a triangulated Klein bottle immersed in R
3
. To generate the dataset
K
, we sampled the following parametrization. Let r
= 41 − cosu2. Then, x
= 6 cos
u1 + sinu + r cosu cosv, if u
π 6 cos
u1 + sinu + r cosv + π, otherwise
y =
16 sin u + r sinu cosv,
if u π
16 sin u,
otherwise z
= r sinv. The underlying space for the other two data sets is the four-dimensional
space-time manifold. For each data set, we triangulate the convex hull of the samples to get a triangulation. Each resulting complex, listed in Table 12.10, is
homeomorphic to a four-dimensional ball and has χ = 1. Data set
E
contains the potential around electrostatic charges at each vertex. Data set
J
records the supersonic flow velocity of a jet engine. We use these values as Morse
functions to generate the filtrations. We then compute persistence over Z
2
coefficients to get the Betti numbers. For each data set, Table 12.10 gives the number s
k
of k-simplices, as well as the Euler characteristic χ = ∑
k
−1
k
s
k
. We use the Morse function to compute the excursion set filtration for each data
set. Table 12.11 gives information on the resulting filtrations.
12.3.3 Field Coefficients
With the generalized algorithm, we may compute the homology of the Klein bottle over different coefficient fields. Here, we are interested only in the Betti
numbers of the final complex in the filtration for illustrative purposes. The nonorientability of this surface is visible in Figure 12.6. The change in tri-
angle orientation at the parametrization boundary leads to a rendering artifact where two sets of triangles are front-facing. In homology, the nonorientabil-
Table 12.10. Data sets. K
is the Klein bottle, shown in Figure 12.6. E
is the potential around electrostatic charges.
J is supersonic jet flow.
number s
k
of k-simplices 1
2 3
4 χ
K 2,000
6,000 4,000
E 3,095
52,285 177,067
212,327 84,451
1 J
17,862 297,372
1,010,203 1,217,319
486,627 1
Table 12.11. Filtrations. The number of simplices in the filtration |K| = ∑
i
s
i
, the length of the filtration number of distinct values of function f , time to
compute the filtration, and time to compute persistence over Z
2
coefficients.
|K| len
filt s pers s
K 12,000
1,020 0.03
0.01 E
529,225 3,013
3.17 5.00
J 3,029,383
256 24.13
50.23
ity manifests itself as a torsional 1-cycle c where 2c is a boundary indeed, it bounds the surface itself. The homology groups over Z are
H
K = Z,
H
1
K = Z × Z
2
,
H
2
K = {0}. Note that
β
1
= rank
H
1
= 1. We now use the “height function” as our Morse function, f
= z, to generate the filtration in Table 12.11. We then compute the homology of data set
K
with field coefficients using our algorithm, as shown in Table 12.12.
Over Z
2
, we get β
1
= 2 as homology is unable to recognize the torsional boundary 2c with coefficients 0 and 1. Instead, it observes an additional class
of homology 1-cycles. By the Euler-Poincaré relation, χ = ∑
i
β
i
, so we also get a class of 2-cycles to compensate for the increase in
β
1
. Therefore, Z
2
homology misidentifies the Klein bottle as the torus. Over any other field, however, homology turns the torsional cycle into a boundary, as the inverse
of 2 exists. In other words, while we cannot observe torsion in computing homology over fields, we can deduce its existence by comparing our results
