The Vietoris–Rips Complexes of FiniteSubsets of an Ellipse of Small Eccentricity

Samadwara ReddySubmitted in completion of the requirements

for Graduation with Distinction in Mathematics

Duke UniversityDurham, North Carolina

2017

1 Introduction

Let X be a metric space and let r > 0 be a scale parameter. The Vietoris–Rips simplicial complex VR<(X; r) (resp. VR≤(X; r)) has X as its vertexset and has a finite σ ⊆ X as a simplex whenever the diameter of σ is lessthan r (resp. at most r). When we do not want to distinguish between <and ≤, we use the notation VR(X; r) to denote either complex. Vietoris–Rips complexes have been used to define a cohomology theory for metricspaces [14], in geometric group theory [9], and more recently in computationaltopology and topological data analysis [8, 6].

In the setting of computational topology, metric space X is often a fi-nite sample from an unknown space M , and one would like to use X torecover information about M . A common technique is to use the thicken-ing VR(X; r) as an approximation for M , although it is not clear how tochoose scale parameter r. The idea of persistent homology is to study nestedsequences of topological spaces, and to trust homological features that per-sist across several spaces as represenative of the true homological featuresof the underlying manifold; transient features are disregarded as noise. Onecommon nested sequence of spaces is the Vietoris-Rips filtration, a sequenceof Vietoris-Rips complexes for increasing scale parameter. There are severaltheoretical results which justify this approach:

• If M is a Riemannian manifold and scale parameter r is sufficientlysmall, then [11, Theorem 3.5] proves that VR<(X; r) is homotopyequivalent to M .

• If M is a Riemannian manifold, scale parameter r is sufficiently small,and metric space X is sufficiently close to M in the Gromov-Hausdorffdistance1, then [13, Theorem 1.1] proves that VR<(X; r) is homotopyequivalent to M .

• If M ⊆ Rd is compact with sufficiently large weak feature size and finiteset X ⊆ Rd is sufficiently close to M in the Hausdorff distance, then [7,Theorem 3.6] proves that the homology groups of M can be recoveredfrom the persistent homology of VR<(X; r) at sufficiently small scaleparameters.

1One way X can be sufficiently close to M in the Gromov-Hausdorff distance is ifX ⊆M is a sufficiently dense.

1

Persistent homology is frequently used for topological data analysis, inwhich a finite sample X from an unknown manifold M is used to studythe topology of the underlying manifold. In low dimensions, we can createvisual representations of the data, and take advantage of our natural abil-ity to recognize patterns. However, as dimensions increase, creating usefulsuch representations quickly becomes infeasible, thus necessitating the devel-opment of mathematical techniques. One common problem with persistenthomology in practice is that one does not know the underlying manifold Mand thus cannot ensure that the scale parameters are sufficiently small (soas to be able to take advantage of the above results). As such, practitionersoften allow the scale parameter to range from small to large, often going farbeyond “sufficiently small” values. Hence Vietoris–Rips complexes with largescale parameters arise commonly in applications of persistent homology, eventhough little is known about their behavior.

In [2] this problem is studied in the specific case of S1, the circle. Adamaszekand Adams demonstrate that as r increases, VR<(S1; r) obtains the homo-topy types S1, S3, S5, . . ., until it is finally contractible. Here we meanSk to denote the k-sphere. They also show that for sufficiently dense finitesamples X ⊆ S1, VR(X; r) ' VR(S1; r). To our knowledge this is the onlyconnected non-contractible manifold whose Vietoris–Rips complex homotopytype is known for all scale parameters. This example agrees with a conjectureof Hausmann [11, (3.12)] that for M a compact Riemannian manifold, theconnectivity of VR<(M ; r) is a non-decreasing function of r.

In this paper we study Vietoris–Rips complexes of ellipses equipped withthe Euclidean metric. We restrict our attention to ellipses Y = {(x, y) ∈R2 | (x/a)2 + y2 = 1} with semi-major axis length a satisfying 1 < a ≤

√2,

henceforth referred to as ellipses of small eccentricity. The threshhold a ≤√

2guarantees that all metric balls in Y are connected. Our main results are thefollowing, where r1 = 4

√3a

a2+3and r2 = 4

√3a2

3a2+1:

Theorem 6.1. Let Y be an ellipse of small eccentricity. For any sufficientlydense finite sample X ⊆ Y , and r1 < r < r2, we have that VR(X; r) '

∨i S2

for some i ≥ 1.

Theorem 6.2. Let Y be an ellipse of small eccentricity. For any r1 < r < r2,ε > 0, and m ≥ 2, there exists an ε-dense finite subset X ⊆ Y such thatVR(X; r) '

∨m−1 S2.

This shows that even for an arbitrarily dense subset X ⊆ Y , the homotopytypes of VR(X; r) and VR(Y ; r) need not agree. Hence the behavior of

2

0

S1 S2

r1 r2

0

S1V5S2

r1 r2

Figure 1: Schematic diagrams of the persistent homology of VR<(Y ; r) (top)and VR≤(Y ; r) (bottom) in homological dimensions 1 and 2. See Theo-rem 6.3 and Corollary 6.3 for precise statements.

the ellipse is quite different from that of the circle with regard to finiteapproximations. Indeed, any sufficiently dense sample of the circle recoversthe homotopy type of VR<(S1; r) [2, Section 5]. In the particular case whenthe Riemannian manifold is the circle, an approximation result similar to[13, Theorem 1.1] is true not only for small scale parameters r but also forarbitrary scale parameters. However, Theorem 6.2 demonstrates that suchan approximation result does not hold at higher scale for the ellipse, whichis not a Riemannian manifold when equipped with the Euclidean metric.

In an upcoming paper [5] we generalize the approach presented herein tohandle the infinite case, with the following results:

Theorem 6.3. Let 1 < a ≤√

2, r1 = 4√3a

a2+3, and r2 = 4

√3a2

3a2+1. Then

VR<(Y ; r) '

{S1 for 0 < r ≤ r1

S2 for r1 < r ≤ r2

and VR≤(Y ; r) '

S1 for 0 < r < r1

S2 for r = r1 or r2∨5 S2 for r1 < r < r2.

These results are shown in Figure 1.

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Theorem 6.3. The 1-dimensional persistent homology of VR<(Y ; r) (resp.VR≤(S1; r)) consists of a single interval (0, r1] (resp. [0, r1)).

The 2-dimensional persistent homology of VR<(Y ; r) consists of a sin-gle interval (r1, r2]. The 2-dimensional persistent homology of VR≤(S1; r)consists of the interval [r1, r2], as well as a point [r, r] on the diagonal withmultiplicity four for every r1 < r < r2.

In Section 2 we provide preliminary concepts and notation on graphs,simplicial complexes, and Vietoris–Rips complexes. In Section 3 we reviewfinite cyclic graphs from [1, 2, 3, 4], and in Section 4 we introduce finite dy-namical systems and their relation to finite cyclic graphs. Section 5 containsnecessary geometric lemmas about ellipses. Finally, in Section 6 we use thetheory of cyclic dynamics to prove our results about the homotopy types offinite subsets of Vietoris–Rips complexes of small eccentricity ellipses.

2 Preliminaries

Directed graphs

A directed graph is a pair G = (V,E) with V the set of vertices and E ⊆ V ×Vthe set of directed edges, where we require that there are no loops and that noedges are oriented in both directions. The edge (v, w) will also be denotedby v → w. A homomorphism of directed graphs h : G → H is a vertexmap such that for every edge v → w in G, either f(v) = f(w) or there isan edge f(v) → f(w) in H. For a vertex v ∈ V we define the out- andin-neighborhoods

N+(G, v) = {w | v → w}, N−(G, v) = {w | w → v},

as well as their respective closed versions

N+[G, v] = N+(G, v) ∪ {v}, N−[G, v] = N−(G, v) ∪ {v}.

For V ′ ⊆ V we let G[V ′] denote the induced subgraph on the vertex setV ′. An undirected graph is a graph in which the orientations on the edgesare omitted.

4

Conventions for the circle

Let S1 be the circle of unit circumference equipped with the arc-length metric.If p1, p2, . . . , ps ∈ S1 then we write

p1 � p2 � · · · � ps � p1

to denote that p1, . . . , ps appear on S1 in this clockwise order, allowing equal-ity. We may replace pi � pi+1 with pi ≺ pi+1 if in addition pi 6= pi+1. Forp, q ∈ S1 we denote the clockwise distance from p to q by ~d(p, q), and wedenote the closed clockwise arc from p to q 6= p by [p, q]S1 = {z ∈ S1 | p �z � q � p}. Open and half-open arcs are defined similarly and denoted(p, q)S1 , (p, q]S1 , or [p, q)S1 .

For a fixed choice of 0 ∈ S1 each point x ∈ S1 can be identified with thereal number ~d(0, x) ∈ [0, 1), and this will be our coordinate system on S1.

Simplicial Complexes

A Vietoris–Rips complex is an example of a simplicial complex. To definethe simplicial complex, we must first define the simplex, the basic unit uponwhich simplicial complexes are built. Let u0, . . . , uk ∈ Rk+1 be affinely in-dependent, or equivalently, let u1 − u0, . . . , uk − u0 be linearly independent.Then a k-simplex is a set of the form

σ =

{t0u0 + . . . tkuk

∣∣∣∣∣ 0 ≤ i ≤ k, ti ≥ 0,k∑i=0

ti = 1

}.

Alternatively, we may simply let σ be the convex hull of u0, . . . , uk (thesmallest convex set containing the points). Intuitively, we may think ofthe k-simplex as the simplest k dimensional polyhedron: the 0-simplex is avertex, the 1-simplex is an edge, the 2-simplex is a triangle, and the 3-simplexis a tetrahedron.

Notice that the boundary of each simplex is composed of lower dimen-sional simplices: vertices form the boundary of an edge, edges and verticesform the boundary of a triangle, and triangles, edges, and vertices form theboundary of a tetrahedron. We refer to these lower dimensional simplicesas faces of the higher dimensional simplex. More formally, we may definethe faces of a simplex as the convex hull some subset of its vertices (theu0, . . . , uk).

5

We use these simplices as the basic building block for more complicatedtopological structures, called simplicial complexes.

Definition 2.1. A geometric simplicial complex is the union of a finite setK of simplices with the property that for any two simplices σ, τ ∈ K, eitherσ ∩ τ = ∅ (they are disjoint) or σ ∩ τ is a simplex of K.

Sometimes it is helpful to abstract from this geometric definition, so herewe provide a purely combinatorial definition of the simplical complex.

Definition 2.2. An abstract simplicial complex is a set K of nonemptyfinite subsets of some finite set V , with the property that for every σ ∈ K,and every nonempty τ ⊆ σ, we also have τ ∈ K.

Upon first examination this definition appears substantially different fromthe geometric definition provided above. This relationship becomes clearerwhen we associate a geometric simplicial complex with abstract simplicialcomplex, and vice-versa. For a given geometric simplicial complex K, wedefine its associated abstract simplicial complex K∗ as

K∗ = {σ | σ is the set of all vertices of some simplex in K}.

Since every simplex in K∗ contains its faces, we see that K∗ has the requiredsubset property to be an abstract simplicial complex. Now for a given ab-stract simplicial complex K∗ on the vertex set V with |V | = n, we relabel Vwith affinely independent points u1, . . . , un ∈ Rn and let

K =⋃

[ui0 ,...,uik ]∈K∗

conv(ui0 , . . . , uik)

where conv(u0, . . . , uk) denotes the convex hull of u0, . . . , uk. We shall hence-forth use the term simplicial complex to refer interchangeably to either anabstract simplicial complex or to its geometric realization.

Vietoris–Rips Complexes

We now define the simplicial complex of interest to this paper, the Vietoris–Rips complex. However, we must first define the Vietoris–Rips graph, towhich the Vietoris–Rips complex is closely related.

6

Figure 2: The blue balls are of radius r, the scale-parameter. Two pointsare connected if they are within r of each other. Finally we take the cliquecomplex of the graph, to arrive at the simplicial complex pictured.

Definition 2.3. Given a metric space X and a real scale parameter r >0, the Vietoris–Rips graph VR<(X; r) (respectively VR≤(X; r)) is the graphwhose vertices are the points in X, and which contains edges between verticesu, v ∈ X if and only if d(u, v) < r (respectively d(u, v) ≤ r).

The definition of the Vietoris–Rips complex also depends upon the notionof a clique within graphs.

Definition 2.4. A clique of a graph G = (V,E) is a subset of vertices C ⊆ Vsuch that for every u, v ∈ C, (u, v) ∈ E.

Definition 2.5. A clique complex of a graph is the (abstract) simplicialcomplex on vertex set V with a simplex for every subset of vertices forminga clique.

Definition 2.6. The Vietoris–Rips complex VR<(X; r) (respectively VR≤(X; r))is the clique complex of VR<(X; r) (respectively VR≤(X; r)), where X is ametric space and r > 0 is a real scale parameter are as above.

When statements are true for a fixed choice of either subscript (< or≤), we will denote the Vietoris–Rips graph and complex as VR(X; r) andVR(X; r), respectively. An example of a Vietoris–Rips complex is picturedin Figure 2

The particular aspect of Vietoris–Rips complexes that we will study ishomotopy type.

7

Homotopy

Definition 2.7. Two continuous functions f, g : X → Y are homotopic ifthere exists a continuous function h : X×[0, 1]→ Y such that h(x, 0) = f(x),and h(x, 1) = g(x) for any x ∈ X.

Interpreting the second parameter of h as time, we may view h as acontinuous deformation from f to g. This intuition carries into the definitionof homotopy equivalent spaces:

Definition 2.8. Two spaces X and Y are homotopy equivalent (denoted ')if there exist continuous functions f : X → Y and g : Y → X such that f ◦ gis homotopic to idY and g ◦ f is homotopic to idX .

This corresponds to the notion that X can be continuously deformed(bent, shrunk, or stretched) into Y , and vice-versa. In this paper we willexpress the homotopy types of the various Vietoris–Rips complexes in termsof their homotopy equivalence to spheres (ex. S5) or wedge sums of spheres(ex. S2 ∨ S2), where the wedge sum is a one point union (glue the spherestogether at a single point).

For a more detailed coverage of topology and combinatorial topology, werefer the reader to Hatcher [10] and Kozlov [12].

3 Cyclic graphs

In this section we recall finite cyclic graphs as studied in [2], and we alsoextend the definitions to include possibly infinite cyclic graphs. Our motiva-tion is that if Y is an ellipse of small eccentricity and X ⊆ Y , then VR(X; r)is a cyclic graph.

The following definition generalizes [2, Definition 3.1] to the possibly in-finite case.

Definition 3.1. A directed graph with vertex set V ⊆ S1 is cyclic if wheneverthere is a directed edge v → u, then there are directed edges v → w and w → ufor all v ≺ w ≺ u ≺ v.

In Figure 3 is pictured an example such graph.We also generalize cyclic graphs homomorphisms ([2, Definition 3.5]) to

the possibly infinite case.

8

0

12

3

4 5

Figure 3: A cyclic graph with a dominated vertex (4).

Definition 3.2. Let G and H be cyclic graphs. A homomorphism f : G→ Hof directed graphs is cyclic if

• whenever v ≺ w ≺ u ≺ v in G, then f(v) � f(w) � f(u) � f(v) in H,and

• f is not constant whenever G has a directed cycle of edges.

Definition 3.3 ([2]). For integers n and k with 0 ≤ k < 12n, the directed

graph Ckn has vertex set {0, . . . , n − 1} and edges i → (i + s) mod n for all

i = 0, . . . , n− 1 and s = 1, . . . , k. Equivalently,

i→ j ⇐⇒ 0 < ~dn(i, j) < k

The graph C39 is pictured in Figure 4. The graphs Ck

n will be useful in ourstudy of Vietoris–Rips graphs. An important numerical invariant of a cyclicgraph is the winding fraction, given for the finite case in [2, Definition 3.7].

Definition 3.4. The winding fraction of a finite cyclic graph G is

wf(G) = sup{kn

: there exists a cyclic homomorphism Ckn → G

}.

9

0

1

23

4

5

67

8

Figure 4: The cyclic graph C39 which has 9 vertices, each of out-degree 3.

10

Note that the winding fraction is necessarily rational.If cyclic graph G is finite, then we often label the vertices in cyclic order

v0 ≺ v1 ≺ . . . ≺ vn−1 ≺ v0.

Definition 3.5 ([2]). A vertex vi in a finite cyclic graph G is dominated (byvi+1) if N−(G, vi+1) = N−[G, vi].

In Figure 4, for example, vertex 4 is dominated by vertex 5.We may dismantle a finite cyclic graph by removing dominated vertices

in rounds as follows. Let S0 be the set of all dominated vertices in G, andin round 0 remove these vertices to obtain G[V \ S0]. Inductively for i > 0,let Si be the set of all dominated vertices in G[V \ (S0 ∪ · · · ∪ Si−1)], and inround i remove these vertices to obtain G[V \ (S0 ∪ · · · ∪ Si)]. We refer to Sias the set of vertices dominated in the i-th round. By [2, Proposition 3.12]every finite cyclic graph dismantles down to an induced subgraph of the formCkn for some 0 ≤ k < 1

2n.

4 Finite Cyclic Dynamical Systems

Associated to each finite cyclic graph is a finite cyclic dynamical system, asstudied in [4].

Definition 4.1. Let G be a finite cyclic graph with vertex set V . The associ-ated finite cyclic dynamical system is generated by the dynamics f : V → V ,where f(v) is defined to be the clockwise most vertex of N+[G, v].

A vertex v ∈ V is periodic if f i(v) = v for some i ≥ 1. If v is periodicthen we refer to {f i(v) | i ≥ 0} as a periodic orbit, whose length ` is thesmallest i ≥ 1 such that f i(v) = v. The winding number of a periodic

orbit v1, v2, . . . , vq, denoted wn(v1, v2, . . . , vq) is∑`−1

i=0~d(f i(v), f i+1(v)), the

number of times that the orbit wraps around the graph. It follows from [4,Lemma 2.3] that every periodic orbit has the same length `(G) and the samewinding number wn(G). We define orb(G) to be the number of periodicorbits.

By [2, Proposition 3.12] any cyclic graph dismantles down to an inducedsubgraph of the form Ck

n. Since the proof of [4, Lemma 3.4] holds in the caseof arbitrary cyclic graphs, rather than solely for Vietoris–Rips graphs of thecircle, we see that that the vertex set of this induced subgraph is the set ofperiodic vertices of f .

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An important numerical invariant of a cyclic graph is the winding fraction,which is originally defined using homomorphisms of cyclic graphs. It followsfrom [2, Proposition 3.14] that the winding fraction of a cyclic graph G canbe equivalently defined as wf(G) = k

n, where G dismantles to Ck

n.

Thus we have wf(G) = kn

= wn(G)`(G)

, so the winding number of G is deter-mined by the dynamical system f . We can describe the homotopy type ofthe clique complex of G via the dynamical system f as follows.

Proposition 4.2. If G is a finite cyclic graph, then

Cl(G) '

{S2l+1 if l

2l+1< wf(G) < l+1

2l+3for some l ∈ N,∨orb(G)−1 S2l if wf(G) = l

2l+1.

Proof. Theorem 4.4 of [2] states

Cl(G) '

{S2l+1 if l

2l+1< wf(G) < l+1

2l+3for some l ∈ N,∨n−2k−1 S2l if wf(G) = l

2l+1.

Suppose G dismantles to Ckn and

l

2l + 1= wf(G) =

k

n. (3)

Then as required we have

orb(G) = gcd(n, k)

=n

2l + 1by (3) since gcd(l, 2l + 1) = 1

= n

(1− 2l

2l + 1

)= n− 2k.

5 Geometric lemmas for the ellipse

We next prove several geometric lemmas about ellipses. These lemmas willallow us to describe the infinite cyclic dynamical systems corresponding

12

to Vietoris–Rips complexes of ellipses, and hence the homotopy types ofVietoris–Rips complexes of ellipses (in Section 6).

Let d : R2 × R2 → R be the Euclidean metric. Let Y = {(x, y) ∈R2 | (x/a)2+y2 = 1} be an ellipse of small eccentricity, meaning the length ofthe semi-major axis satisfies 1 < a ≤

√2. The threshhold a ≤

√2 guarantees

that all metric balls in Y are connected (Lemma 5.5). Note we equip theellipse Y with the Euclidean metric d, even though we have been using (and[2] uses) the geodesic metric on the circle S1.

We will often think Y as S1, simply by fixing an orientation-preservinghomeomorphism Y → S1. We write p1 � p2 � . . . � pn � p1 if p1, p2, . . . , pn ∈Y are oriented in a clockwise fashion, allowing equality. We may also replacepi � pi+1 with pi ≺ pi+1 if in addition pi 6= pi+1. For p, q ∈ Y with p 6= q wedenote the closed clockwise arc from p to q by [p, q]Y = {z ∈ Y | p � z �q � p}. Open and half-open arcs are defined similarly and denoted (p, q)Y ,(p, q]Y , and [p, q)Y .

Definition 5.1. Define the continuous function h : Y → Y which maps apoint p ∈ Y to the unique point in the intersection of Y \{p} with the normalline to Y at p.

Lemma 5.2. Given a point p ∈ Y with p /∈ {(±a, 0), (0,±1)}, there isexactly one point q in the diametrically opposite quadrant (on the other sideof both axes) satisfying h(q) = p. In particular h : Y → Y is surjective.

Proof. Without loss of generality, let p = (x, y) ∈ Y be in the bottom leftquadrant, meaning x < 0 and y < 0. We must find some q = (a cos t, sin t)with 0 < t < π

2and h(q) = p. The direction of the vector perpendicular to

the ellipse at q is (1, a tan t)T , and the direction of the vector from p to q is(a cos t− x, sin t− y)T . Thus h(q) = p whenever

c

(a cos t− xsin t− y

)=

(1

a tan t

)for some c ∈ R, or alternatively, when

g(t) := (a2 − 1) sin t− ax tan t+ y = 0.

The derivative g′(t) = (a2−1) cos t−ax sec2 t is strictly positive on 0 < t < π2

since a > 1 and x < 0. Therefore g(t) is strictly increasing on 0 < t < π2,

13

and so there is at most one q with h(q) = p. To see that there is at least onesuch q, note that g is continuous with g(0) = y < 0 and limt→π

2− g(t) =∞.

To see that h : Y → Y is surjective, note h((±a, 0)) = (∓a, 0) andh((0,±1)) = (0,∓1).

Lemma 5.3. Ellipse Y = {(x, y) ∈ R2 | (x/a)2 + y2 = 1} with a > 1 is anellipse of small eccentricity (meaning a ≤

√2) if and only if for each point

q ∈ Y not on an axis, h(q) is in the quadrant diametrically opposite from q.

Proof. We restrict attention, without loss of generality, to the case whereq = (a cos t, sin t) with 0 < t < π/2. Let L be the line perpendicular to Yat q. The slope of L is a tan t, so the intersection of L with the x-axis is thepoint (0, y0) where

y0 = sin t− (a cos t)(a tan t) = (1− a2) sin t.

For 1 < a ≤√

2 we have −1 < y0 < 0, and hence the h(q) must lie inthe lower left quadrant. For a >

√2 we see that for t sufficiently close to

π/2, we’ll have y0 < −1, meaning that point h(q) will be in the lower rightquadrant and the property does not hold.

Lemma 5.4. The function h is bijective if and only if Y is an ellipse ofsmall eccentricity.

Proof. If Y is an ellipse of small eccentricity, then h is bijective by Lemmas 5.2and 5.3. If Y is not an ellipse of small eccentricity, then by Lemmas 5.3 and5.2 there is some p ∈ Y with points q in an adjacent quadrant and q′ in theopposite quadrant satisfying h(q) = p = h(q′). Hence h is not injective.

For p ∈ Y , define the function dp : Y → R by dp(q) = d(p, q).

Lemma 5.5. For Y an ellipse of small eccentricity and p ∈ Y , the onlycritical points of dp : Y → R are a global minimum at p and a global maxi-mum at h−1(p). It follows that nonempty metric balls in Y are either con-tractible or all of Y . Furthermore, dp is continuous and strictly increasingon (p, h−1(p))Y .

Proof. Clearly p is the global minimum of dp. For any critical point q 6= p ofdp it must be the case that the circle of radius d(p, q) centered at p is tangentto Y at q, and hence h(q) = p. By Lemma 5.4, h is bijective, and since Y iscompact it follows that h−1(p) is the global maximum of dp and that thereare no other critical points besides p and h−1(p).

14

Let Y be an ellipse of small eccentricity. If p ∈ Y and 0 ≤ r ≤ dp(h−1(p)),

then we define gr(p) ∈ Y to be the unique solution q to dp(q) = r on[p, h−1(p)]Y . Uniqueness and existence are guaranteed by the fact that dpis continuous and strictly increasing on the interval. It is not hard to seethat gr : Y → Y is bijective and of degree (i.e. winding number) one. Itfollows that

Lemma 5.6. For Y an ellipse of small eccentricity and p ∈ Y , the function(0, 2)→ Y defined by r 7→ gr(p) is continuous.

Proof. Function dp : (p, h−1(p))Y → R is continuous since (p, h−1(p))Y is theimage of the continuous curve in the plane. Hence given ε > 0, there existsδ > 0 such that d(q, gr(p)) < δ implies |dp(q) − r| < ε. Choose q− ∈(p, gr(p))Y ∩Bmin(δ,ε)(gr(p)), and q+ ∈ (gr(p), h

−1(p))Y ∩Bmin(δ,ε)(gr(p)); theseintersections are nonempty since (p, h−1(p))Y is a continuous curve throughgr(p). By construction, d(gr(p), q

−), d(gr(p), q+) < ε. Since dp is strictly

increasing, as we move along Y from p to h−1(p), by Lemma 5.5, we havedp(q

−) < r < dp(q+). Thus for |r − r′| < min(r − dp(q−), dp(q

+)− r), we seethat gr′(p) ∈ (q−, q+)Y and hence d(gr′(p), gr(p)) < ε.

Lemma 5.7. For Y an ellipse of small eccentricity and 0 ≤ r < 2, thefunction gr : Y → Y is continuous.

Proof. Let ε > 0 and p ∈ Y . By Lemma 5.6, there exists some δ > 0such that |r − r′| ≤ δ implies d(gr(p), gr′(p)) < ε; choose such a δ withδ < ε. By the triangle inequality, we see that if p′ satisfies d(p′, p) < δ, thend(p′, gr−δ(p)) < r < d(p′, gr+δ(p)). Hence, by continuity of d, there is somepoint q′ in the interval (gr−δ(p), gr+δ(p))Y ⊆ Bε(gr(p)) satisfying d(p′, q′) = r,and thus gr : Y → Y is continuous.

For X ⊆ Y and 0 ≤ r < 2, we orient the edges of VR≤(X; r) (resp.VR<(X; r)) in a clockwise fashion by specifying that p → p′ is a directededge if p′ ∈ (p, gr(p)]Y (resp. p′ ∈ (p, gr(p))Y ).

Lemma 5.8. Let Y be an ellipse of small eccentricity, let X ⊆ Y , and let0 ≤ r < 2. Then VR≤(X; r) and VR<(X; r) are cyclic graphs.

Proof. We can think of the vertex set X as a subset of S1, simply by fixingan orientation-preserving homeomorphism Y → S1. If p → p′ is a directededge of VR≤(X; r), then p′ ∈ (p, gr(p)]Y . If p′′ ∈ X satisfies p ≺ p′′ ≺ p′ ≺ p,

15

In[711]:= Demo1[]

Out[711]=

Angle thetatheta 1.1

Semimajor axis lengtha 1.3

Equilateral triangle side lengthsidelength 1.92337

p

q

q′

6 EquilateralTriangleInEllipse_Groebner_GraphicsEquilateralExists.nb

Figure 5: Proof of Lemma 5.9

then we have p′′ ∈ (p, gr(p)]Y by definition and p′ ∈ (p′′, gr(p′′)]Y since gr is

cyclic. Hence we have directed edges p → p′′ and p′′ → p′ in VR≤(X; r),and therefore VR≤(X; r) is a cyclic graph. The proof that VR<(X; r) is acyclic graph is identical after replacing each half-open interval with an openinterval.

Lemma 5.9. Let Y be an ellipse of small eccentricity and let p ∈ Y . Thenthere exists a unique inscribed equilateral triangle in Y containing y as oneof its vertices.

Proof. Rotate Y by 60◦ clockwise about y. Clearly there must be at leastsome intersection point between the two ellipses other than y. This inter-section point corresponds to two points p and q on the original ellipse. Byconstruction, we see that yp = yq and that ∠pyq = 60◦, so we must have anequilateral triangle.

For uniqueness, suppose for a contradiction that {p, gr(p), g2r(p)} and{p, gr′(p), g2r′(p)} are two distinct inscribed equilateral triangles in Y with sidelengths r < r′ (we will learn later in Theorem 5.12 that necessarily r′ < 2,but this isn’t needed now). Necessarily we have gr(p), gr′(p) ∈ (p, h−1(p))Yand g2r(p), g

2r′(p) ∈ (h−1(p), p)Y . Note gr(p) ∈ (p, gr′(p))Y implies, since gr is

a bijection of degree one,

g2r(p) ∈ (gr(p), gr(gr′(p)))Y ⊆ (gr(p), g2r′(p))Y .

This gives p ≺ h−1(p) ≺ g2r(p) ≺ g2r′(p) ≺ p. Hence r = d(g2r(p), p) >d(g2r′(p), p) = r′, a contradiction.

Remark 5.10. The proof of Lemma 5.9 relies on a ≤√

2 and Lemmas 5.4

and 5.5. Simulations show that Lemma 5.9 is true also for a ≤√

73.

16

For Y an ellipse of small eccentricity, by Lemma 5.9 we may define acontinuous function s : Y → R which maps each y ∈ Y to the side length ofthe unique inscribed equilateral triangle containing this vertex.

Lemma 5.11. For Y an ellipse of small eccentricity, the function s : Y → Ris continuous.

Proof. By the fact that dp is strictly increasing along this interval, we see thatgr(p) is strictly increasing in r, in the sense that, for r < r′ < d(p, h−1(p)) < 2,we have

p ≺ gr(p) ≺ gr′(p) ≺ h−1(p) ≺ p.

Therefore,

gr′(p) ≺ gr′(gr(p)) ≺ g2r′(p) ≺ gr′(h−1(p)) ≺ gr′(p)

and thus for |r − r∗| sufficiently small,

gr(p) ≺ g2r(p) ≺ g2r′(p) ≺ gr(h−1(p)) ≺ gr(p)

and alsog2r(p) ≺ g3r(p) ≺ g3r′(p) ≺ g2r(h

−1(p)) ≺ g2r(p).

Let ε > 0, p, p∗ ∈ Y , and r = s(p). Without loss of generality supposethat p ≺ p∗ ≺ gr(p) ≺ p. Choose r+, r−, such that

r − ε < r− < r < r+ < r + ε

andg2r(p) ≺ g3r−(p) ≺ p ≺ g3r+(p) ≺ g2r(h

−1(p)) ≺ g2r(p).

Then by the continuity and monotonicity of g3r , if d(p, p∗) is sufficiently small,

g2r(p) ≺ g3r−(p∗) ≺ p∗ ≺ g3r+(p∗) ≺ g2r(h−1(p)) ≺ g2r(p).

By continuity of g3r , there exists r∗ such that r− < r∗ < r+ and g3r∗(p∗) = p∗.

Then, r∗ = s(p∗) and |r − r∗| < ε so s is continuous.

One can verify that

∆(a,0) ={

(a, 0),(−3a− a3

a2 + 3,

2√

3a

a2 + 3

),(−3a− a3

a2 + 3,− 2√

3a

a2 + 3

)}∆(−a,0) =

{(−a, 0),

(3a− a3

a2 + 3,

2√

3a

a2 + 3

),(3a− a3

a2 + 3,− 2√

3a

a2 + 3

)}17

are inscribed equilateral triangles of side length r1 = s(±a, 0) = 4√3a

a2+3, and

that

∆(0,1) ={

(0, 1),( 2√

3a2

3a2 + 1,−3a2 − 1

3a2 + 1

),(− 2√

3a2

3a2 + 1,−3a2 − 1

3a2 + 1

)}∆(0,−1) =

{(0,−1),

( 2√

3a2

3a2 + 1,3a2 − 1

3a2 + 1

),(− 2√

3a2

3a2 + 1,3a2 − 1

3a2 + 1

)}are inscribed equilateral triangles of side length r2 = s(0,±1) = 4

√3a2

3a2+1. Fur-

thermore we have r1 < r2 < 2.

Theorem 5.12. For Y an ellipse of small eccentricity, the six vertices inthe two inscribed equilateral triangles ∆(±a,0) are global minima of s : Y → R,and the six vertices in the two inscribed equilateral triangles ∆(0,±1) are globalmaxima. There are no other local extrema of s.

Proof. We first show that given any r > 0, there are at most twelve pointsp ∈ Y satisfying s(p) = r. Let (x1, y1), (x2, y2) ∈ Y be two points in theellipse at distance r apart. We consider all points (x, y) ∈ R2 at distancer from each of (x1, y1) and (x2, y2). The triangle {(x1, y1), (x2, y2), (x, y)} isthen an inscribed equilateral triangle in Y if and only if (x, y) ∈ Y .

Consider the following system of polynomial equations in x1, y1, x2, y2, x, y, a, r ∈R:

(x1/a)2 + y21 − 1 = 0

(x2/a)2 + y22 − 1 = 0

(x/a)2 + y2 − 1 = 0 (13)

(x1 − x2)2 + (y1 − y2)2 − r2 = 0

(x− x1)2 + (y − y1)2 − r2 = 0

(x− x2)2 + (y − y2)2 − r2 = 0.

The first three equations encode the requirement that (x1, y1), (x2, y2), (x, y) ∈Y , and the last three equations encode the requirement that {(x1, y1), (x2, y2), (x, y)}is an equilateral triangle. One can use elimination theory to eliminate thevariables x1, y1, x2, y2, y from (13), producing a polynomial p(x, a, r) such thatany solution to (13) is also a root of p. We use the mathematics software

18

system Sage to compute

p(x, a, r) = 12288a12r4 − 6912a12r6 − 10752a10r6 + 13568a8r6 + 1296a12r8

+ 4032a10r8 − 2208a8r8 − 6720a6r8 + 3600a4r8 − 81a12r10

− 378a10r10 − 63a8r10 + 1044a6r10 − 63a4r10 − 378a2r10 − 81r10

− 36864a10r4x2 + 36864a8r4x2 + 13824a10r6x2 − 4608a8r6x2

+ 16896a6r6x2 − 26112a4r6x2 − 1296a10r8x2 − 432a8r8x2

− 4512a6r8x2 + 4512a4r8x2 + 432a2r8x2 + 1296r8x2 + 36864a8r4x4

− 73728a6r4x4 + 36864a4r4x4 − 6912a8r6x4 + 15360a6r6x4

− 16896a4r6x4 + 15360a2r6x4 − 6912r6x4 − 12288a6r4x6

+ 36864a4r4x6 − 36864a2r4x6 + 12288r4x6.

Note p is a polynomial in x of degree 6. It follows that given any a andr, a solution to the system of equations (13) can contain at most 6 distinctvalues for x, and hence at most 12 distinct points p = (x, y) ∈ Y satisfyings(p) = r.

We next show that the extrema of s : Y → R are as claimed. Symmetryof s about the horizontal and vertical axes, along with the fact that thecardinality of each set s−1(r) is finite2, gives that the points (±a, 0), (0,±1)are local extrema of s. Since s(p) = s(gs(p)(p)) = s(g2s(p)(p)), it follows thatall twelve vertices in the triangles ∆(±a,0),∆(0,±1) are local extrema. Sincethe six vertices of ∆(±a,0) are interleaved with the six vertices of ∆(0,±1), itfollows from the intermediate value theorem that for all r1 < r < r2 wehave |s−1(r)| ≥ 12. Since |s−1(r)| ≤ 12 for all r > 0, it follows that sis either monotonically increasing or monotonically decreasing between anytwo adjacent vertices of the triangles ∆(±a,0) and ∆(0,±1). Hence the verticesof ∆(±a,0) are global minima, the vertices of ∆(0,±1) are global maxima, andthere are no other local extrema of s.

6 Vietoris–Rips complexes of ellipses of small

eccentricity

We restrict attention in this section to Y an ellipse of small eccentricity.

2This rules out infinite oscillations such as in the topologists’ sine curve

19

��������� ��

��

EquilateralTriangleInEllipse_Groebner_Graphics.nb 9

Figure 6: The blue equilateral triangles have side length r1 and the redequilateral triangles have side length r2, where r1 < r2. Theorem 5.12 saysthat r1 and r2 are the smallest and largest values of s : Y → R.

For finite X ⊆ Y it follows from Lemma 5.8 that VR(X; r) is homotopyequivalent to an odd sphere or a wedge of even spheres of the same dimension.

The following notation will prove convenient. Given a point p ∈ Y , letp′ = gs(p)(p) and let p′′ = g2s(p)(p). By Theorem 5.12 and the intermediatevalue theorem, there are cyclically ordered points

z0 ≺ z1 ≺ z2 ≺ z3 ≺ z′0 ≺ z′1 ≺ z′2 ≺ z′3 ≺ z′′0 ≺ z′′1 ≺ z′′2 ≺ z′′3 ≺ z0

in Y such that

• s(z0) = s(z1) = s(z2) = s(z3) = r.

• I0 = (z0, z1)Y ∪ (z′0, z′1)Y ∪ (z′′0 , z

′′1 )Y and I2 = (z2, z3)Y ∪ (z′2, z

′3)Y ∪

(z′′2 , z′′3 )Y are each sets of three equivalence classes of fast points for

dynamics gr : Y → Y , meaning that s(p) < r for all points p in theseintervals.

• I1 = (z1, z2)Y ∪ (z′1, z′2)Y ∪ (z′′1 , z

′′2 )Y and I3 = (z3, z

′0)Y ∪ (z′3, z

′′0 )Y ∪

(z′′3 , z0)Y are each sets of three equivalence classes of slow points fordynamics gr : Y → Y , meaning that s(p) > r for all points p in theseintervals.

Theorem 6.1. Let Y be an ellipse of small eccentricity. For any sufficientlydense finite sample X ⊆ Y , the graph VR(X; r) for r1 < r < r2 has at least

20

two periodic orbits of length three, and hence VR(X; r) '∨i S2 for some

i ≥ 1.

Proof. We cyclically order the elements of X as p0 ≺ p1 ≺ . . . ≺ pn−1 ≺ p0,where all arithmetic operations on vertex indices are performed modulo n.

Recall from Definition 4.1 that associated to the cyclic graph VR(X; r)we have a map f : X → X. Since fm converges pointwise to gmr as thedensity increases, for X sufficiently dense each interval

(z0, z2)Y , (z2, z′0)Y , (z

′0, z′2)Y , (z

′2, z′′0 )Y , (z

′′0 , z′′2 )Y , (z

′′2 , z0)Y (1)

contains points pi ∈ (z0, z1)Y , and pj ∈ (z1, z2)Y such that

pi ≺ f 3(pi) � f 3(pj) ≺ pj ≺ pi. (2)

Now, suppose for a contradiction that there is no point x ∈ [pi, pj]Y ∩Xwith x = f 3(x). Since f 3 weakly preserves the cyclic ordering we have

pi ≺ f 3(pi) � f 3(pi+1) � f 3(pj) ≺ pj ≺ pi,

and since pi+1 6= f 3(pi+1) this gives

pi+1 ≺ f 3(pi+1) � f 3(pj) ≺ pj ≺ pi+1.

Note this is (2) with i replaced by i+ 1. Iterating this process gives

pj−1 ≺ f 3(pj−1) � f 3(pj) ≺ pj ≺ pj−1,

contradicting the fact that (pj−1, pj)Y ∩ X = ∅. Hence there must be somex ∈ [pi, pj]Y ∩X with x = f 3(x). Repeating this argument with each intervalin (1) gives 6 periodic points, and hence at least two periodic orbits of lengththree.

The statement about the homotopy type of VR(X; r) follows from Propo-sition 4.2.

Theorem 6.2. Let Y be an ellipse of small eccentricity. For any r1 < r < r2,ε > 0, and m ≥ 2, there exists an ε-dense finite subset X ⊆ Y so thatthe graph VR(X; r) has exactly m periodic orbits of length three, and henceVR(X; r) '

∨m−1 S2.

21

Proof. We will prove the theorem by showing that for any n, n′ ≥ 1 withn + n′ = m, one can produce an ε-dense subset X of Y \ {zi, z′i, z′′i | i =0, 1, 2, 3} such that I0 ∩ X contains n periodic orbits for VR(X; r), I2 ∩ Xcontains n′ periodic orbits for VR(X; r), and I1 ∩X and I3 ∩X contain noperiodic orbits for fr.

Pick a point p1 ∈ (z0, z1)Y such that d(p1, z1), d(p′1, z′1), d(p′′1, z

′′1 ) < ε. Add

the points p1, p′1, p′′1 to X; these points will form our first periodic orbit in

I0 ∩X. To see this, note that since p1 is a fast point, s(p1) < r, so:

p′1 = gs(p)(p1) ≺ gr(p1) ≺ gr(z1) = z′1 ≺ p′1

So long as we don’t add any points in (p′1, gr(p1)]Y , (p′′1, gr(p′1)]Y , or (p1, gr(p

′′1)]Y ,

we have that fr(p1) = p′1, fr(p′1) = p′′1, and fr(p

′′1) = p1, and hence a peri-

odic orbit of length 3. Next pick a point p2 satisfying p2 ∈ (gr(p′′1), z1)Y ,

p′2 ∈ (gr(p1), z′1)Y , and p′′2 ∈ (gr(p

′1), z

′′1 )Y . Add the points p2, p

′2, p

′′2 to X to

form our second periodic orbit. Iterating, for 2 ≤ i ≤ n we pick a point pisatisfying pi ∈ (gr(p

′′i−1), z1)Y , p′i ∈ (gr(pi−1), z

′1)Y , and p′′i ∈ (gr(p

′i−1), z

′′1 )Y ,

and we add the points pi, p′i, p′′i to X to form our i-th periodic orbit. This

creates n periodic orbits in I0 ∩X.We now create ε-density in I0. First note that by the continuity of gs(p)(p)

in p, that for p ∈ (z0, z1)Y , and q ∈ (z0, p)Y sufficiently close to p,

d(q, p), d(q′, p′), d(q′′, p′′) < ε

Furthermore, for any point p ∈ (z0, z1)Y , since p is fast, d(p′, gr(p)) > 0.Hence, by continuity of gr, we see that for q ∈ (z0, p)Y sufficiently close to p:

q′ ≺ p′ ≺ gr(q) ≺ gr(p) ≺ q′

Combining these facts: for any p ∈ (z0, z1)Y , and q ∈ (z0, p)Y sufficientlyclose to p, we have both of the above sets of conditions. Let u ∈ (z0, z1)Ysatisfy d(z0, u) < ε/4. By compactness of [u, p1]Y , we have that there existsδ′ > 0 such that for all p ∈ [u, p1]Y , and q ∈ (z0, p)Y , with d(p, q) < δ′,the above conditions hold. Let δ = min(δ′, ε)/2. Add q1 = g−1δ (p1), q

′1,

q′′1 to X. Note that this doesn’t create any periodic orbits. To see this,simply observe that p′1 ≺ gr(q

′1) ≺ gr(p1) ≺ p′2 ≺ p′1, and hence fr(q1) = p′1.

Inductively define qn = g−1δ (qn−1), for n > 1. Iteratively add qn, q′n, q′′n to

X, so long as qn ∈ [u, p1]Y . Notice similarly that this does not create anyadditional periodic orbits. Since δ > 0, this process will terminate with someqn. Notice that d(qn, z0) ≤ d(qn, u) + d(u, z) < 3

4ε, so we have ε-density in I0.

22

Create n′ periodic orbits in an ε-dense subset of I2 via an analogousprocedure.

Finally, we make X ε-dense by adding otherwise arbitrary points to I1 andI3. These new points cannot create any new periodic orbits for fr because I1and I3 are sets of slow points for gr : Y → Y . This concludes the constructionof an ε-dense subset X so that VR(X; r) has exactly m periodic orbits.

The statement about the homotopy type of VR(X; r) follows from Propo-sition 4.2.

In an upcoming paper [5] [Note to the committee: The preprint will beupon arXiv by the time I submit the final draft of my thesis, and will be citedhere] we generalize the cyclic dynamic approach presented herein to handlethe infinite case, with the following results:

Theorem 6.3. Let 1 < a ≤√

2, r1 = 4√3a

a2+3, and r2 = 4

√3a2

3a2+1. Then

VR<(Y ; r) '

{S1 for 0 < r ≤ r1

S2 for r1 < r ≤ r2

and VR≤(Y ; r) '

S1 for 0 < r < r1

S2 for r = r1 or r2∨5 S2 for r1 < r < r2.

Furthermore,

• For 0 < r < r′ ≤ r1 or r1 < r < r′ ≤ r2, inclusion VR<(Y ; r) ↪→VR<(Y ; r′) is a homotopy equivalence.

• For 0 < r < r′ < r1, inclusion VR≤(Y ; r) ↪→ VR≤(Y ; r′) is a homo-topy equivalence.

• For r1 ≤ r < r′ ≤ r2, inclusion VR≤(Y ; r) ↪→ VR≤(Y ; r′) induces arank 1 map on 2-dimensional homology H2( ;F) for any field F.

Corollary 6.3. The 1-dimensional persistent homology of VR<(Y ; r) (resp.VR≤(S1; r)) consists of a single interval (0, r1] (resp. [0, r1)).

The 2-dimensional persistent homology of VR<(Y ; r) consists of a sin-gle interval (r1, r2]. The 2-dimensional persistent homology of VR≤(S1; r)consists of the interval [r1, r2], as well as a point [r, r] on the diagonal withmultiplicity four for every r1 < r < r2.

23

7 Future work

In the future, we hope to expand on this work two primary ways. First wewish to expand to the case of r2 < r < 2, so as to more fully understand thehomotopy types of small eccentricity ellipses. Secondly, we hope to study thecase of ellipses of larger eccentricity a >

√2. In this case, we are no longer

guaranteed cyclic graphs, and must therefore resort to alternative techniquesto study homotopy type.

8 Acknowledgements

I would like to thank Henry Adams for his invaluable mentorship and guid-ance. This research would not have been possible without the support of Dr.Kraines and the PRUV Fellowship. Furthermore, my thanks to Justin Curryfor his co-mentoring the project during the PRUV Fellowship, and Paul Ben-dich for acting as the Duke University faculty supervising my independentstudy. I would also like to thank Michal Adamaszek for his assistance withsome of the proofs. Lastly, I would like to thank the committee for takingthe time to review this paper.

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[3] Micha l Adamaszek, Henry Adams, Florian Frick, Chris Peterson, andCorrine Previte-Johnson. Nerve complexes of circular arcs. Discrete &Computational Geometry, 56:251–273, 2016.

[4] Micha l Adamaszek, Henry Adams, and Francis Motta. Random cyclicdynamical systems. Advances in Applied Mathematics, 83:1–23, 2017.

[5] Micha l Adamaszek, Henry Adams, and Samadwara Reddy. On Vietoris–Rips complexes of ellipses. Preprint.

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[6] Gunnar Carlsson. Topology and data. Bulletin of the American Math-ematical Society, 46(2):255–308, 2009.

[7] Frederic Chazal and Steve Oudot. Towards persistence-based recon-struction in Euclidean spaces. In Proceedings of the 24th Annual Sym-posium on Computational Geometry, pages 232–241. ACM, 2008.

[8] Herbert Edelsbrunner and John L Harer. Computational Topology: AnIntroduction. American Mathematical Society, Providence, 2010.

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[11] Jean-Claude Hausmann. On the Vietoris–Rips complexes and a co-homology theory for metric spaces. Annals of Mathematics Studies,138:175–188, 1995.

[12] Dmitry N Kozlov. Combinatorial Algebraic Topology, volume 21 of Al-gorithms and Computation in Mathematics. Springer, 2008.

[13] Janko Latschev. Vietoris–Rips complexes of metric spaces near a closedRiemannian manifold. Archiv der Mathematik, 77(6):522–528, 2001.

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