Vietoris–Rips and Restricted C̆ech Complexes of Circular Points
1
2
3
Michał Adamaszek , Henry Adams , Chris Peterson , and Corrine Previte-Johnson
1
2
3
3
Max Planck Institute for Informatics, Institute for Mathematics and its Applications, and Colorado State University
Main Theorem
Case of evenly-spaced points
Theorem. A Vietoris–Rips complex or a restricted C̆ech
complex on a finite set of points from the circle is homotopy
equivalent to either a point, an odd sphere, or a wedge sum
of spheres of the same even dimension.
Corollary 6.7 from [1].
Let k < n/2 and write n − k = q(n − 2k) + r with 0 ≤ r < n − 2k.
Remark: The homotopy types of Vietoris–Rips complexes of
evenly-spaced circular points are proven in [1].
k=1
1
n=6 S
7
S1
1
8
S
9
S1
1
10
S
11
S1
1
12
S
13
S1
1
14
S
15
S1
16
S1
17
S1
18
S1
19
S1
20
S1
1
21
S
22
S1
1
23
S
24
S1
1
25
S
26
S1
1
27
S
28
S1
Remark: The proof for arbitrary circular points relies on the
evenly-spaced case, which we consider first.
Notation for evenly-spaced points
Definition. Let VR(n, k) be the Vietoris–Rips complex on
n evenly-spaced circular vertices with connectivity parameter
2πk/n. That is, simplex σ is in VR(n, k) when diam(σ) ≤
2πk/n.
Example. VR(9, 3) is the clique complex of the graph below,
giving VR(9, 3) ' ∨2S 2.
To visualize this, note that VR(9, 3) has three maximal 2simplices.
Definition. Let restricted C̆ech complex C̆(n, k) be the
nerve of the covering of S 1 by n evenly-spaced closed arcs
of arc length 2πk/n.
Example. C̆(6, 3) is the nerve of the 6 circular arcs below,
giving C̆(6, 3) ' ∨2S 2.




∨n−2k−1S
T hen VR(n, k) '  2q−1

S
2
3
2
S
∗
S1
∗
1
3
S
S
S 1 ∨2S 2
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
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
4
∗
∗
∗
∗
4
S
S3
2
∨3S
S1
1
S
S1
S1
S1
S1
S1
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
5
∗
∗
∗
∗
∗
∗
5
S
S3
3
S
∨4S 2
S1
S1
S1
S1
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
6
∗
∗
∗
∗
∗
∗
∗
∗
6
S
∨2S 4
S3
S3
∨5S 2
S1
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
2q−2
7
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S7
S5
S3
S3
S3
2
∨6S
S1
1
S
S1
1
S
S1
1
S
S1




∨n−k−1S
T hen C̆(n, k) '  2q−1

S
if r = 0
otherwise.
8
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S8
S5
∨3S 4
3
S
S3
3
S
∨7S 2
1
S
S1
1
S
S1
9
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S9
6
∨2S
S5
3
S
S3
3
S
S3
2
∨8S
S1
10
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S 10
7
S
S5
4
∨4S
S3
3
S
S3
11
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S 11
7
S
S5
5
S
S3
12
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
S 12
8
∨2S
∨3S 6
2q−2
if r = 0
otherwise.
The proof uses [1, 2, 3].
k=1
1
n=3 S
4
S1
1
5
S
6
S1
7
S1
8
S1
9
S1
10
S1
11
S1
1
12
S
13
S1
1
14
S
15
S1
1
16
S
17
S1
1
18
S
19
S1
1
20
S
21
S1
1
22
S
23
S1
24
S1
2
3
∗
∗
S2
∗
1
3
S
S
S 1 ∨2S 2
S1 S1
S1 S1
S1 S1
S1 S1
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
4
∗
∗
∗
S4
S3
∨3S 2
S1
S1
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
S1
5
∗
∗
∗
∗
S5
S3
S3
∨4S 2
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
S1
6
∗
∗
∗
∗
∗
S6
∨2S 4
S3
S3
2
∨5S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
S1
7
∗
∗
∗
∗
∗
∗
S7
S5
S3
3
S
S3
2
∨6S
S1
1
S
S1
1
S
S1
1
S
S1
1
S
S1
S1
8
∗
∗
∗
∗
∗
∗
∗
S8
S5
4
∨3S
S3
3
S
S3
2
∨7S
S1
1
S
S1
1
S
S1
1
S
S1
S1
9
∗
∗
∗
∗
∗
∗
∗
∗
S9
6
∨2S
S5
3
S
S3
3
S
S3
2
∨8S
S1
1
S
S1
1
S
S1
S1
10
11
12
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
10
S
∗
∗
S 7 S 11
∗
5
7
12
S
S
S
∨4S 4 S 5 ∨2S 8
3
5
6
S
S
∨3S
S3
S3
S5
3
3
4
S
S
∨5S
S3
S3
S3
2
3
3
∨9S S
S
S1
S3
S3
1
2
3
S ∨10S S
S1
S1
S3
S1
S 1 ∨11S 2
Remark: Note VR(n + k, k) ' C̆(n, k). For example, VR(9, 3) ' ∨2S 2 ' C̆(6, 3).
Arbitrary circular points
Alternatively, C̆(6, 3) has 6 maximal 3-simplices, glued together to form C̆(6, 3) ' ∨2S 2.
Proposition.
Let k < n − 1 and write n = q(n − k) + r with 0 ≤ r < n − k.
When built on an arbitrary finite set of circular points, a Vietoris–Rips
or restricted C̆ech complex is still homotopy equivalent to either a point,
an odd sphere, or a wedge sum of spheres of the same even dimension.
Proof idea: If K is a simplicial complex and u and v are two distinct
vertices with st(u) ⊆ st(v) (we say u is dominated by v), then K '
K \ u. We show how to remove dominated vertices until we are left with
a complex equivalent to some VR(n, k) or C̆(n, k).
References
[1] Michał Adamaszek, Clique complexes and graph powers, Israel
Journal of Mathematics 196 (2013), 295–319.
[2] Jonathan Barmak, Star clusters in independence complexes of
graphs, Advances in Mathematics 241 (2013), 33–57.
[3] Jakob Jonsson On the topology of independence complexes of
triangle-free graphs, unpublished manuscript.