The Vietoris-Rips Complex of the Circle
1
2
Michał Adamaszek and Henry Adams
1
2
Max Planck Institute for Informatics, and Institute for Mathematics and its Applications
Homotopy Types of VR(X, r) for X ⊂ S
Notation
1
1
1
Let S be the circle of unit circumference with the geodesic metric.
1
Definition. For X ⊂ S and r ≥ 0, let VR(X, r) be the Vietoris–Rips complex with
vertex set X and connectivity parameter r. That is, simplex σ ⊂ X is in VR(X, r) when
diam(σ) ≤ r.
Theorem 2. For X ⊂ S finite, VR(X, r) is homotopy equivalent to either a point,
an odd sphere, or a wedge sum of spheres of the same even dimension.
1
For example, let Xn ⊂ S consist of n evenly-spaced points. This case is given in [1].
2
Example. We have VR(X9, 1/3) ' ∨2S .
1
Persistent Homology of VR(S , r)
1
Theorem 1. The odd-dimensional persistent homology of VR(S , r) is






l
l
+
1
1 

 ,
dgm H2l+1 VR(S ) = 
,

2l + 1 2l + 3
1
2l+1
and within this interval VR(S , r) ' S . The even-dimensional persistent homology
has no intervals of positive length.
H3
H1
0
1/3
H5
2/5 3/7 ...
References
[1] Michał Adamaszek, Clique complexes and graph powers, Israel Journal of Mathematics 196 (2013),
295–319.
[2] Frédéric Chazal, Vin de Silva, Steve Oudot, Persistence stability for geometric complexes, Geometriae
Dedicata (2013), 1–22.
[3] Jean-Claude Hausmann, On the Vietoris–Rips complexes and a cohomology theory for metric spaces,
Annals of Mathematics Studies 138 (1995), 175–188.
[4] Janko Latschev, Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold,
Archiv der Mathematik 77 (2001), 522–528.
Corollary 6.7 from [1]. Let k < n/2. Then





∨n−2k−1S
2l
VR(Xn, k/n) '  2l+1

S
k=1
1
n=4 S
5
S1
6
S1
7
S1
8
S1
9
S1
10
S1
1
11
S
12
S1
1
13
S
14
S1
1
15
S
16
S1
1
17
S
18
S1
1
19
S
20
S1
1
21
S
22
S1
1
23
S
24
S1
25
S1
26
S1
27
S1
28
S1
2
3
∗
∗
∗
∗
S2
∗
S1
∗
S1 S3
S 1 ∨2S 2
S1 S1
1
1
S
S
S1 S1
1
1
S
S
S1 S1
1
1
S
S
S1 S1
1
1
S
S
S1 S1
1
1
S
S
S1 S1
1
1
S
S
S1 S1
1
1
S
S
S1 S1
S1 S1
S1 S1
S1 S1
S1 S1
4
∗
∗
∗
∗
∗
∗
S4
3
S
∨3S 2
1
S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
S1
S1
S1
S1
if
if
5
∗
∗
∗
∗
∗
∗
∗
∗
S5
3
S
S3
2
∨4S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
S1
S1
S1
S1
k
n
l
= 2l+1
l
k
<
2l+1
n
6
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S6
4
∨2S
S3
3
S
∨5S 2
1
S
S1
1
S
S1
1
S
S1
S1
S1
S1
S1
7
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S7
5
S
S3
3
S
S3
2
∨6S
S1
1
S
S1
S1
S1
S1
S1
<
8
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S8
5
S
∨3S 4
3
S
S3
3
S
∨7S 2
S1
S1
S1
S1
l+1
2l+3
9
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S9
6
∨2S
S5
3
S
S3
S3
S3
∨8S 2
S1
for some l ≥ 0.
10
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S 10
7
S
S5
∨4S 4
S3
S3
S3
11
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S 11
S7
S5
S5
S3
12
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S 12
∨2S 8
∨3S 6
13
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S 13