@ Appl. Gen. Topol. 20, no. 1 (2019), 251-264doi:10.4995/agt.2019.10928
c© AGT, UPV, 2019

Exact computation for existence of a

knot counterexample

K. Marinelli and T. J. Peters

Department of Computer Science & Engineering, University of Connecticut, USA

(kevin.marinelli@uconn.edu,tpeters@cse.uconn.edu)

Communicated by M. Schellekens

Abstract

Previously, numerical evidence was presented of a self-intersecting

Bézier curve having the unknot for its control polygon. This numerical

demonstration resolved open questions in scientific visualization, but

did not provide a formal proof of self-intersection. An example with

a formal existence proof is given, even while the exact self-intersection

point remains undetermined.

2010 MSC: 68U05.

Keywords: knot theory; isotopy; parametric curve.

1. Introduction to Existence Condition for Self-intersection

The formal proof of an unknotted control polygon with a self-intersecting
Bézier curve (See Definition 3.3.) appears in Theorem 8.4. The progression in
Figure 1, from left to right, shows ‘snap shots’ of a piecewise linear (PL) curve
being linearly perturbed, generating new Bézier curves at each instant. The
top-most point is the only one perturbed. The initial (left) and final (right)
Bézier curves are shown to have differing prime knot types, so there must be
an intermediate Bézier curve with a self-intersection (as shown in the middle
image by a black dot). Relevant definitions follow.

Received 25 April 2018 – Accepted 06 November 2018

http://dx.doi.org/10.4995/agt.2019.10928


K. Marinelli and T. J. Peters

Figure 1. Existence of self-intersection

2. Data for the Example

Let

(0, 9, 20), (−15, −95, −50), (40, 80, −20), (−10, −60, 58),

(−60, 30, 20), (40, −60, −60), (0, 9, 20)

be the ordered list of vertices for the initial PL knot, denoted by K0. Integer
values are chosen for exact computation. In Figure 1, the vertex (0, 9, 20) is
the initial and final point of the Bézier curves (shown as the highest point
on the polynomial curve, colored green). The remaining vertices are arranged
counterclockwise. A single Reidemeister move of vertex (40, -60, -60) shows
that K0 is the unknot.

The vertex in bold font is perturbed linearly to (10, −60, 58) to obtain
the final PL knot, denoted by K1, with all other vertices remaining fixed. The
perturbation generates uncountably many new control polygons and associated
Bézier curves. The Bézier curves corresponding to K0 and K1 are denoted by
β0 and β1, respectively. Both K0 and K1 resulted from visual experiments.

3. Mathematical Preliminaries

Some relevant mathematical definitions are summarized.

Definition 3.1. A subspace of R3 is called a knot [7] if it is homeomorphic to
a circle.

The stronger equivalence of ambient isotopy is fundamental in knot theory.

Definition 3.2. Let X and Y be two subspaces of R3. A continuous function

H : R3 × [0, 1] → R3

is an ambient isotopy between X and Y if H satisfies the following conditions:

(1) H(·, 0) is the identity,
(2) H(X, 1) = Y , and
(3) ∀t ∈ [0, 1], H(·, t) is a homeomorphism from R3 onto R3.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 252



Computational knot counterexample

Definition 3.3. Denote C(t) as the parameterized Bézier curve of degree n
with control points Pi ∈ R

3, defined by

C(t) =

n
∑

i=0

Bi,n(t)Pi, t ∈ [0, 1],

where Bi,n(t) =

(

n
i

)

ti(1 − t)n−i.

The curve P formed by PL interpolation on the ordered list of points

P0, P1, . . . , Pn

is called the control polygon and is a PL approximation of C.
The work presented was partially motivated by development of wireless data

gloves. The smooth Bézier representations are manipulated by the gloves, while
the illustrative computer graphics are created by PL approximations derived
from the control polygon. Topological fidelity between these representations is
of interest to provide reliable visual feedback to the user.

4. Related Work

This article arose from a question posed by a referee1 from a previous nu-
merical example [22].

The mathematical objects of study here are Bézier curves. The canonical ref-
erences [13] (any edition) and [27] focus on approximation and modeling, with
less attention to associated topological properties. Relations between a smooth
curve and its PL approximation are dominant in computer graphics [14], but
the topological aspects are often ignored. One prominent property is that any
Bézier curve is contained in the convex hull of its control points [13], while re-
cent enhancements have been shown [26], where the containing set is a subset
of the convex hull.

The push receives prominent attention by R. H. Bing [8] as a fundamental
tool in developing ambient isotopies in ℜ2. The perturbation in this article is
the trivial extension of a push to ℜ3.

The preservation of topological characteristics in geometric modeling and
graphics has become of contemporary interest [4, 5, 6, 10, 12, 16, 19, 21]. Suf-
ficient conditions for a homeomorphism between a Bézier curve and its control
polygon have been studied [25], while topological differences have also been
shown [9, 22, 27].

Topological aspects are relevant in ‘molecular movies’ [23]. Sufficient con-
ditions were established for preservation of knot type [17] during dynamic vi-
sualization of ongoing molecular simulations [29]. Interest by bio-chemists in
using hand gestures to interactively manipulate these complex molecular im-
ages prompts related topological considerations for data gloves [1, 2, 3].

1Author T. J. Peters thanks that referee of the previous numerical work [22] for a sug-
gestion that led to the example shown here.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 253



K. Marinelli and T. J. Peters

The beautiful visual studies of knot symmetry [28] provided an example of
the 74 knot that was helpful in the initial visual studies performed to create
the example presented here.

Exact computation has some advantages over floating point arithmetic [11,
18], particularly regarding loss of precision in finite bit strings, with alternative
views expressed [15]. Some ad hoc techniques are often invoked to provide
adequate bit length for full precision relative to a particular computation, as
has been done here.

5. Exact Computation for Subdivision

The de Casteljau algorithm [13] is a subdivision method for Bézier curves2.
The algorithm recursively generates control polygons that more closely approx-
imate the curve [13] under Hausdorff distance [24]. The algorithm is presented
here, for a Bézier curve denoted by α, for the input value of t = 1/2, as is
customary3. The point α(1/2), is an endpoint output by the algorithm. For
t = 1/2 subdivision proceeds by selecting the midpoint of each edge of the con-
trol polygon and these midpoints are connected to create new edges, as shown
in the left image of Figure 2. Recursive creation and connection of midpoints
continues until an edge is created that is tangent to the Bézier curve [13].
Termination is guaranteed since there are only finitely many edges.

This splits the original curve into two pieces, at α(1/2), designated here as
the ‘left’ and ‘right’ pieces of α. The left piece has a control polygon with a
final point of α(1/2) and the right piece has a control polygon with initial point
of α(1/2), as shown in the right image of Figure 2, where α(1/2), is the point of
tangency for terminating the algorithm. The union of the edges from the final
step then forms two new PL curves (each described as a sub-control polygon4)
as shown in blue and green in the right image of Figure 2.

Figure 2. A subdivision of a cubic Bézier curve at parameter 1
2

For each iteration, m, the subdivision process generates 2m PL sub-curves,
each being a sub-control polygon [13], as denoted by Pmj for j = 1, 2, 3, . . . , 2

m.
For a Bézier curve initially defined by n + 1 control points, each Pmj has n + 1

points and their union
⋃

j
Pmj forms a new PL curve that converges in Hausdorff

distance to the original Bézier curve. The Bézier curve defined by
⋃

j
Pmj is

exactly the same as the original Bézier curve [20].

2An illustrative image appears [13, Figure 3.2].
3Any parameter t from (0, 1) can be substituted.
4Note that each sub-control polygon of a simple Bézier curve is open.

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Computational knot counterexample

Remark 5.1. A Bézier curve is contained within the convex hull of its control
points [13].

Remark 5.1 also applies to each sub-control polygon, as will be extensively
used. The stick knots of Section 2 are refined via the DeCasteljau subdivision
algorithm, repeatedly computing averages for the coordinate vertices in the

form
(a+b)

2
. In order to retain integer valued vertices throughout all computed

subdivisions, the vertices will be scaled by 2k, for an appropriately chosen
k. The existence of self-intersections in the Bézier curves is preserved under
scaling. Each subdivision iteration has one division by 2 for each degree d. In
our examples, we invoke ℓ levels of subdivision, so that k = ℓ ∗ (d + 1) + 1.

6. Ambient Isotopy Between Stick Knots

The two curves, K0 and K1 will be shown to be the unknot.

Lemma 6.1. The curve K0 is the unknot.

Proof. The demonstration that K0 is simple proceeds directly over the six
edges. The non-consecutive pairs of edges have separating hyperplanes. �

Lemma 6.2. The curves K0 and K1 are ambient isotopic.

Proof. The single perturbation to create K1 is similar to a push [8] and incurs
no self-intersections with other segments. It is easily extended to an ambient
isotopy having compact support. �

Corollary 6.3. The curve K1 is the unknot.

7. Subdivision Analysis

The points from the 4th iteration of the DeCasteljau algorithm, using a
subdivision value of 1/2 are listed in Appendix A. These can be verified simply
by executing the algorithm on the initial data for K0 and K1, with the provision
that the implementation relies upon exact arithmetic.

It was observed that each sub-control polygon in Appendix A had the prop-
erty that its control points were strictly monotone in one of its coordinate
values. This is annotated in the appendix by abbreviations noting the coordi-
nates in which strict monotonicity was observed. In some cases, this occurs for
all three coordinates. This observation greatly simplified the proofs presented.

Let K0,4 denote the PL curve formed from the 4th subdivision on K0 and
similarly denote K1,4 for K1. Within each of K0,4 and K1,4, there are 16 sub-
control polygons. The rest of the paper proceeds by showing that K0,4 is the
unknot and is ambient isotopic to β0 and that K1,4 is the trefoil and is ambient
isotopic to β1. As the convex hulls of the sub-control polygons become central,
their images are shown in Figure 3.

Lemma 7.1. Both K0,4 and K1,4 are simple.

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K. Marinelli and T. J. Peters

(a) K0 (b) K1

Figure 3. Convex Hulls – Fourth subdivision (parameter 1
2
)

Proof. First consider K0,4. Let K0,4,i, for i = 0, 1, . . .15, denote the 16 sub-
control polygons for K0,4. Each K0,4,i, for i = 0, 1, . . . 15, is simple, because of
the strict monotonicity in at least one coordinate. The K0,4,i, for i = 0, 1, . . .15,
are pairwise disjoint (except at shared endpoints), as established by computing
the convex hull for each. Again, any ambiguity that could arise from floating
point arithmetic is avoided by exact computations.

The argument for K1,4 follows the same pattern. �

The derivative of a Bézier curve C is expressed as

C
′

(t) =

n
∑

i=0

(

n

i

)

ti(1 − t)n−i(Pi+1 − Pi), i ∈ {1, . . . , n − 1}.

Lemma 7.2. Each of β0 and β1 are strictly monotonic in the same coordinate
as their corresponding control polygons and are simple.

Proof. The strict monotonicity in some coordinate for each sub-control poly-
gon implies that the partial derivative in the x, y or z variable will be positive,
implying simplicity for that segment of the Bézier curve. Since each Bézier seg-
ment is contained in the convex hull of its sub-control polygon and since those
convex hulls are pairwise disjoint (except at shared endpoints), the simplicity
follows. �

Corollary 7.3. If a control polygon is strictly monotonic in some coordinate,

then its associated Bézier curve is also.

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Computational knot counterexample

Subdivision need not preserve knot type, so we consider K0,4 and K1,4.

Lemma 7.4. The curve K0,4 is the unknot.

Proof. Project K0,4 onto the X − Y plane, where 3 intersections are computed
exactly, listed according to the oriented direction of K0,4, with the z coordinates
also computed exactly.

Overcrossing at (x, y) =

(70600874219532518400/90998355737, −331936500725210686464/90998355737),

since z: 346821834258400839680/90998355737 > −179855360,

Overcrossing at (x, y) =

(−2606810091676070400/2228701007, −141298996572704411136/15600907049),

since z: −695629074309606400/821100371 > −2734161920,

Undercrossing at (x, y) =

(70736463317966883840/46441845451, −2183313901438239630336/232209227255),

since z: −312474348049921062912/46441845451 < −5234410880.

This crossing information permits the conclusion that K0,4 is the unknot. �

Lemma 7.5. The curve K1,4 is the trefoil.

Proof. Project K1,4 onto the X − Y plane, as done, similarly, in the proof of
Lemma 7.4, with, again, 3 intersections computed exactly.

Undercrossing at (x, y) =

(−2321511588927897600/2778711187, −12860064917297823744/2778711187),

since z: = 8352902508791726080/2778711187 < 4026531840,

Overcrossing at (x, y) =

(−8217249783839948800/7079707793, −58835463893254373376/7079707793),

since z: = −24693447116718080/7079707793 > −1547779858,

Undercrossing at (x, y) =

(2736710937161450496/968030497, −222045650166834551808/24200762425),

since z: = −6443291820066594816/968030497 < −5348303360.

These three pairs of alternating crossings permit conclusion of the trefoil. �

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 257



K. Marinelli and T. J. Peters

8. Ambient Isotopy for Bézier Curves

We show that β0 is ambient isotopic to K0,4 and that β1 is ambient isotopic
to K1,4. Both proofs proceed by showing that each sub-control polygon is
ambient isotopic to its associated Bézier curve. We show this for one sub-
control polygon, as representative. Let K0,4,i be one such control polygon and
let β0,i be its corresponding Bézier curve segment.

Theorem 8.1. Each K0,4,i is ambient isotopic to its corresponding Bézier
curve, denoted as β0,i.

Proof. Without loss of generality assume K0,4,i is strictly monotonic in its x
coordinate, which implies that β0,i is also strictly monotonic in its x coordinate
by Corollary 7.3. For each p ∈ K0,4,i, let Πp be the plane at p that is parallel
to the Y Z-plane. By the indicated strict monotonicity, Πp ∩ K0,4,i = {p},
is a unique point. Similarly, by the same strict monotonicity on β0,i, the
intersection Πp ∩β0,i has at most one point, and the connectivity of β0,i (as the
continuous image of a connected subset of [0, 1]) implies that this intersection
is non-empty. For each p ∈ K0,4,i, let qp = Πp ∩ β0,i. To construct the ambient
isotopy, consider the line segment between each p and qp. The interiors of these
line segments will not intersect, due to the strict monotonicity. The remaining
details to construct an ambient isotopy of compact support are standard and
left to the reader. �

Corollary 8.2. The Bézier curve β0 is ambient isotopic to its control polygon
K0,4, the unknot.

Proof. The existence of an ambient isotopy within each convex hull of K0,4,i,
the lack of intersection between convex hulls K0,4,i and K0,4,j for i 6= j (except
at the end points when j = i + 1, where those end points remain fixed) and
composition of the ambient isotopies on each sub-control polygon conclude this
proof. �

Corollary 8.3. The Bézier curve β1 is ambient isotopic to its control polygon
K1,4, the trefoil.

Theorem 8.4. The ambient isotopy between K0 and K1 generates a Bézier
curve with a self-intersection.

Proof. The perturbation of the control point (−10, 60, 58) to (10, 60, 58) creates
a homotopy on the Bézier curve β0 to β1, defined over the interval [0, 1]. If
for all values of t ∈ [0, 1], the perturbed Bézier curve was simple, then the
homotopy would be an isotopy between β0 and β1. However, since β0 and β1
have differing prime knot types, this cannot be. So, for some value t̃ ∈ [0, 1],
the corresponding Bézier curve must have a self-intersection. �

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Computational knot counterexample

9. Conclusion and Future Work

An example is shown of a closed Bézier curve and its control polygon which
are both the unknot. A perturbation of one vertex is shown to cause the per-
turbed Bézier curve to be the trefoil, while its control polygon remains the
unknot. This transition demonstrates that there must have been an intermedi-
ate state where a self-intersection occurred in the transforming Bézier curves.
These topological differences between the mathematical representation and its
PL approximation are of interest in computer graphics and animation. Vi-
sual evidence previously existed, but the only supportive mathematics relied
on floating point arithmetic which left open the question of whether an inter-
mediate intersection could be rigorously proven. The formal proof presented
relies on the implementation of exact, integer computations to resolve that
open question.

Comparisons between the exact techniques and certifiable numeric methods
merit further consideration, as it may often be of interest to specify a neighbor-
hood in which a self-intersection is known to exist. The present example was
quite carefully constructed to show the self-intersection and robust numerical
methods are likely to have broader scope for applications.

Acknowledgements. The authors acknowledge, with appreciation, the con-

tributions of

• D. Marsh, for software that generated experimental visualizations and
related computations,

• the reviewers, for singularly comprehensive and constructive comments,
and

• the editors, for their keen insight and informed perspective in selecting
those reviewers.

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Computational knot counterexample

Appendix A. Subdivision Data

The following two tables list the vertices for both the original and perturbed
control polygons, each after 4 subdivisions. The strict monotonicity of specific
coordinates is indicated for each subdivided control polygon.

In Table I for subdivision of K0, the sixteen control polygons are denoted
by P[0], . . . , P[15] and all consecutive control polygons are separated by half-
planes. Between P[0] and P[1] the separating plane is the XZ plane at the
y-value of the last point of P[0]. Between P[1] and P[2] the separating plane is
the XZ plane at the y-value of the last point of P[1]. For all others, it is a Y Z
plane at the x value of the shared subdivision point.

In Table II for subdivision of K1, the sixteen control polygons are denoted
by Q[0], . . . , Q[15] and all consecutive control polygons are separated by half-
planes. Between Q[9] and Q[10] the separating plane is the XZ plane at the
y-value of the last point of P[0]. Between Q[14] and Q[15] the separating plane
is the XZ plane at the y-value of the last point of P[14]. For all others, it is a
Y Z plane at the x value of the shared subdivision point.

A notational change for points occurs in Appendix A. In the preceding nar-
rative, points are denoted by parentheses in the format of (x, y, z). Within
Appendix A, points are bounded by the set brackets in the format of {x, y, z},
as the required syntax within Mathematica, where many of the computations
were performed. For integrity of the data, it was judged best to report the
points using this alternative formatting.

The code used and its output is posted for public access at

https://github.com/kmarinelli/AGT 2018 TP KM

Table I: Subdivision Points from 4 Iterations on K0

P[0]: Y,Z

{ 0, 4831838208,10737418240 },

{ -503316480, 1342177280,8388608000 },

{ -859832320, -1562378240,6249512960 },

{ -1092485120, -3959685120,4313317376 },

{ -1221918720, -5918269440,2572288000 },

{ -1266603520, -7498398720,1017953280 },

{ -1242964800, -8753032512,-358732160 }

P[1]: X,Y,Z

{ -1242964800, -8753032512,-358732160},

{ -1219326080, -10007666304,-1735417600},

{ -1127363840, -10936804608,-2934453760},

{ -983503360, -11593406976,-3964886016},

{ -802124800, -12023124992,-4836333568},

{ -595691520, -12265252864,-5558824960},

{ -374886400, -12353556480,-6142648320}

P[2]: X

{ -374886400, -12353556480,-6142648320},

{ -154081280, -12441860096,-6726471680},

{ 81095680, -12376339456,-7171627008},

{ 319961600, -12190760448,-7488402432},

{ 553614080, -11913360640,-7687345664},

{ 774794880, -11567678336,-7779114240},

{ 977745600, -11173260096,-7774340480}

P[3]; X,Y,Z

{ 977745600, -11173260096,-7774340480},

{ 1180696320, -10778841856,-7769566720 },

{ 1365416960, -10335687680,-7668250624 },

{ 1526149120, -9863344128,-7481024512 },

{ 1658634240, -9377366016,-7218495488 },

{ 1759969280, -8890023936,-6891110400 },

{ 1828454400, -8410890240,-6509035520 }

P[4]: Y,Z

{ 1828454400, -8410890240,-6509035520 },

{ 1896939520, -7931756544,-6126960640 },

{ 1932574720, -7460831232,-5690195968 },

{ 1933660160, -7007686656,-5208907776 },

{ 1899699200, -6579196928,-4692981760 },

{ 1831246080, -6180123904,-4151902720 },

{ 1729745600, -5813581632,-3594648960 }

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K. Marinelli and T. J. Peters

P[5]: X,Y,Z

{ 1729745600, -5813581632,-3594648960 },

{ 1628245120, -5447039360,-3037395200 },

{ 1493697280, -5113027840,-2463966720 },

{ 1327546880, -4814661120,-1883341824 },

{ 1132129280, -4553405440,-1303992320 },

{ 910510080, -4329543680,-733777920 },

{ 666316800, -4142518272,-179855360 }

P[6]: X,Y,Z

{ 666316800, -4142518272,-179855360 },

{ 422123520, -3955492864, 374067200 },

{ 155356160, -3805303808, 911697920 },

{ -130357760, -3691393536,1425880064 },

{ -430828800, -3612364032,1910159872 },

{ -741473920, -3566319744,2358877440 },

{ -1057492800, -3551088960,2767242880 }

P[7]: X,Z

{ -1057492800, -3551088960,2767242880 },

{ -1373511680, -3535858176,3175608320 },

{ -1694904320, -3551440896,3543621632 },

{ -2016870400, -3595665408,3866492928 },

{ -2334392320, -3666083840,4140302336 },

{ -2642411520, -3760193536,4362076160 },

{ -2936012800, -3875536896,4529848320 }

P[8]: X,Y

{ -2936012800, -3875536896,4529848320 },

{ -3229614080, -3990880256,4697620480 },

{ -3508797440, -4127457280,4811390976 },

{ -3768647680, -4282810368,4869193728 },

{ -4004392960, -4454526976,4870070272 },

{ -4211589120, -4640339456,4814131200 },

{ -4386312000, -4838103360,4702602880 }

P[9]: Y,Z

{ -4386312000, -4838103360,4702602880 },

{ -4561034880, -5035867264,4591074560 },

{ -4703284480, -5245582592,4423956992 },

{ -4809136640, -5465104896,4202476544 },

{ -4875187200, -5692412928,3928975360 },

{ -4898744320, -5925586944,3606958080 },

{ -4878028800, -6162665472,3241123840 }

P[10]: X,Y,Z

{ -4878028800, -6162665472,3241123840 },

{ -4857313280, -6399744000,2875289600 },

{ -4792325120, -6640727040,2465638400 },

{ -4681285120, -6883653120,2016869376 },

{ -4523326720, -7126519040,1534876160 },

{ -4318696320, -7367136640,1026778880 },

{ -4068961600, -7602868032, 500941440 }

P[11]: X,Y,Z

{ -4068961600, -7602868032, 500941440},

{ -3819226880, -7838599424, -24896000},

{ -3524387840, -8069444608,-568473600},

{ -3186012160, -8292765696,-1121427456},

{ -2806988800, -8505475072,-1674149888},

{ -2391736320, -8703770624,-2215772160},

{ -1946419200, -8882749440,-2734161920}

P[12]: X,Y,Z

{ -1946419200, -8882749440,-2734161920},

{ -1501102080, -9061728256,-3252551680 },

{ -1025720320, -9221390336,-3747708928 },

{ -526438400, -9356832768,-4207501312 },

{ -11166720, -9462051840,-4618532864 },

{ 510222080, -9529556736,-4966141440 },

{ 1025668800, -9549861696,-5234410880 }

P[13]: X

{ 1025668800, -9549861696,-5234410880 },

{ 1541115520, -9570166656,-5502680320 },

{ 2050620160, -9543271680,-5691610624 },

{ 2542123520, -9459691008,-5785285632 },

{ 3001379840, -9307943936,-5766535168 },

{ 3411732480, -9074046976,-5616947200 },

{ 3753881600, -8740884480,-5316894720 }

P[14]: Y,Z

{ 3753881600, -8740884480,-5316894720 },

{ 4096030720, -8407721984,-5016842240 },

{ 4369976320, -7975293952,-4566325248 },

{ 4556418560, -7426484736,-3945716736 },

{ 4633414400, -6741046528,-3134174720 },

{ 4576145280, -5894969984,-2109669120 },

{ 4356676800, -4859733312,-849023360 }

P[15]: X,Y,Z

{ 4356676800, -4859733312,-849023360 },

{ 4137208320, -3824496640, 411622400 },

{ 3755540480, -2600099840,1908408320 },

{ 3183738880, -1158021120,3664510976 },

{ 2390753280, 534773760,5704253440 },

{ 1342177280, 2516582400,8053063680 },

{ 0, 4831838208,10737418240 }

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 262



Computational knot counterexample

Table II: Subdivision Points from 4 Iterations on K1

Q[0]: Y,Z

{ 0, 4831838208,10737418240 },

{ -503316480, 1342177280,8388608000 },

{ -859832320, -1562378240,6249512960 },

{ -1089863680, -3959685120,4312924160 },

{ -1212088320, -5918269440,2570813440 },

{ -1243563520, -7498398720,1014497280 },

{ -1199764800, -8753032512,-365212160 }

Q[1]: X, Y, Z

{ -1199764800, -8753032512,-365212160 },

{-1155966080, -10007666304,-1744921600},

{ -1036893440, -10936804608,-2948024320 },

{ -858022400, -11593406976,-3983708160 },

{ -633246720, -12023124992,-4861665280 },

{ -374917120, -12265252864,-5591941120 },

{ -93900800, -12353556480,-6184796160 }

Q[2]: X

{ -93900800, -12353556480,-6184796160 },

{ 187115520, -12441860096,-6777651200 },

{ 490818560, -12376339456,-7233085440 },

{ 806341120, -12190760448,-7561359360 },

{ 1124299520, -11913360640,-7772948480 },

{ 1436734080, -11567678336,-7878405120 },

{ 1737028800, -11173260096,-7888232960 }

Q[3]: X, Y, Z

{ 1737028800, -11173260096,-7888232960 },

{ 2037323520, -10778841856,-7898060800 },

{ 2325478400, -10335687680,-7812259840 },

{ 2594877440, -9863344128,-7641333760 },

{ 2840248320, -9377366016,-7395737600 },

{ 3057582080, -8890023936,-7085752320 },

{ 3244032000, -8410890240,-6721372160 }

Q[4]: X, Y, Z

{ 3244032000, -8410890240,-6721372160 },

{ 3430481920, -7931756544,-6356992000 },

{ 3586048000, -7460831232,-5938216960 },

{ 3707883520, -7007686656,-5475041280 },

{ 3794304000, -6579196928,-4977172480 },

{ 3844686080, -6180123904,-4453918720 },

{ 3859345600, -5813581632,-3914088960 }

Q[5]: Y, Z

{ 3859345600, -5813581632,-3914088960 },

{ 3874005120, -5447039360,-3374259200 },

{ 3852942080, -5113027840,-2817853440 },

{ 3796472320, -4814661120,-2253680640 },

{ 3705850880, -4553405440,-1690050560 },

{ 3583150080, -4329543680,-1134673920 },

{ 3431116800, -4142518272,-594575360 }

Q[6]: X, Y, Z

{ 3431116800, -4142518272,-594575360 },

{ 3279083520, -3955492864, -54476800 },

{ 3097717760, -3805303808, 470343680 },

{ 2889766400, -3691393536, 972861440 },

{ 2658650880, -3612364032,1446737920 },

{ 2408324480, -3566319744,1886407680 },

{ 2143108800, -3551088960,2287152640 }

Q[7]: X, Z

{ 2143108800, -3551088960,2287152640 },

{ 1877893120, -3535858176,2687897600 },

{ 1597788160, -3551440896,3049717760 },

{ 1307115520, -3595665408,3367895040 },

{ 1010565120, -3666083840,3638558720 },

{ 713031680, -3760193536,3858759680 },

{ 419430400, -3875536896,4026531840 }

Q[8]: X, Y

{ 419430400, -3875536896,4026531840 },

{ 125829120, -3990880256,4194304000 },

{ -163840000, -4127457280,4309647360 },

{ -444661760, -4282810368,4370595840 },

{ -711700480, -4454526976,4376166400 },

{ -960184320, -4640339456,4326420480 },

{ -1185710400, -4838103360,4222512640 }

Q[9]: X, Y, Z

{ -1185710400, -4838103360,4222512640 },

{ -1411236480, -5035867264,4118604800 },

{ -1613804800, -5245582592,3960535040 },

{ -1789012480, -5465104896,3749457920 },

{ -1932825600, -5692412928,3487621120 },

{ -2041784320, -5925586944,3178414080 },

{ -2113228800, -6162665472,2826403840 }

Q[10]: Y, Z

{ -2113228800, -6162665472,2826403840 },

{ -2184673280, -6399744000,2474393600 },

{ -2218603520, -6640727040,2079580160 },

{ -2212359680, -6883653120,1646530560 },

{ -2164081920, -7126519040,1180989440 },

{ -2072936320, -7367136640, 689914880 },

{ -1939361600, -7602868032, 181501440 }

Q[11]: X, Y, Z

{ -1939361600, -7602868032, 181501440 },

{ -1805786880, -7838599424,-326912000 },

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 263



K. Marinelli and T. J. Peters

{ -1629783040, -8069444608,-852664320 },

{ -1411788800, -8292765696,-1387560960 },

{ -1153515520, -8505475072,-1922170880 },

{ -858193920, -8703770624,-2445803520 },

{ -530841600, -8882749440,-2946498560 }

Q[12]: X, Y, Z

{ -530841600, -8882749440,-2946498560 },

{ -203489280, -9061728256,-3447193600 },

{ 155893760, -9221390336,-3924951040 },

{ 542289920, -9356832768,-4367810560 },

{ 948894720, -9462051840,-4762542080 },

{ 1366849280, -9529556736,-5094635520 },

{ 1784952000, -9549861696,-5348303360 }

Q[13]: X

{ 1784952000, -9549861696,-5348303360 },

{ 2203054720, -9570166656,-5601971200 },

{ 2621305600, -9543271680,-5777213440 },

{ 3028503040, -9459691008,-5858242560 },

{ 3411102720, -9307943936,-5827993600 },

{ 3752929280, -9074046976,-5668126720 },

{ 4034867200, -8740884480,-5359042560 }

Q[14]: Y, Z

{ 4034867200, -8740884480,-5359042560 },

{ 4316805120, -8407721984,-5049958400 },

{ 4538854400, -7975293952,-4591656960 },

{ 4681899520, -7426484736,-3964538880 },

{ 4723884800, -6741046528,-3147745280 },

{ 4639505280, -5894969984,-2119173120 },

{ 4399876800, -4859733312,-855503360 }

Q[15]: X, Y, Z

{ 4399876800, -4859733312,-855503360 },

{ 4160248320, -3824496640, 408166400 },

{ 3765370880, -2600099840,1906933760 },

{ 3186360320, -1158021120,3664117760 },

{ 2390753280, 534773760,5704253440 },

{ 1342177280, 2516582400,8053063680 },

{ 0, 4831838208,10737418240 }

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 264