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arXiv:math/0403028v4 [math.GT] 25 Nov 2004
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CHAPTER 1
Minimal Flat Knotted Ribbons
Louis H. Kauffman
Department of Mathematics, Statistics and Computer Science (m/c 249)
851 South Morgan Street
University of Illi nois at Chicago
Chicago, Illinois 60607-7045
E-mail: kauffman@uic.edu
This paper gives mathematical models for flat knotted ribbons, and
makes specific conjectures for the least length of ribbon (for a given
width) needed to tie the trefoil knot and the figure eight knot.
1. Introduction
We study framed knots, configured as knotted r ibbons, where a ribbon is
a s pace homeomor phic to a rectangle. T hink of that rectangle as a long
strip of paper, and imagine tying a knot in the strip and then pulling it
gently tight so that it becomes a flat folded k notted strip. See Figure 3 for
an illustration of this process for the trefoil knot. The folds in the strip
should all be of the type shown in Figure 2. The question about such knots
is this: What is the least length of strip (for a given width) needed to make
a flattened version of the knot?
This question, about length to width ratio for flat knots, is analogous
to a question a bout length to radial width (so-called thickness) for knotted
tubes embedded in three dimensional space. The question for knotted tubes
has a good-sized recent literature
1,2,5,7
, but to this date there is no specific
conjecture for the exact thickness o f even the tr e foil knot. In the present
paper, we make specific conjectures for minimal length to width ratio for
the trefoil and figure eight as knotted flat ribbons. It is clear from this initial
exploration of knotted ribbons that there is much to think about and very
interes ting relationships between geometry and topology in this domain.
1
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2 L. Kauffman
The paper is organized as follows. Section 2 gives our mathematical
model for flat kno tted ribbons, in terms of immersed piecewise linear rib-
bons in the plane. We emphasize two approaches to the length to width
ratio, one for closed knots, and one for truncated k nots. In Section 3 we
analyze the geometry of the folded trefoil and conjecture its minimal ratio
for the truncated trefoil knot. In Section 4, we analyze the ge ometry of
the folded figure eight knot and conjecture its minimal ratio. Section 5 is a
discusssion of possible relationships with fully three dimensional issues.
2. A Mathematical Model for Flat Knotted Ribbons
In Figure 1 we depict two knotted ribbons (A and B) embedded in three
dimensional space. The pr ojection of ribbon A to the plane is not the sort
of flattened knot that we are considering here. The twist in the embedding
flattens to a singular projection that we wish to avoid. The projection of
ribbon B to the plane is suitable for our purpos es.
A
B
Figure 1. Knotted Ribbons
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Minimal Flat Knotted Ribbons 3
Figure 2. Mirrors and Folds
In Figure 2 we provide a mathematical model fo r the sort of folded
strip that we have in mind. In the mathematical model, there is a piece wise
linear curve in the plane with transverse self-intersections. This curve is the
projection of the center-line of the strip to the plane after all folds have been
made. Each fold is modeled as follows: Consider an angle in the piecewise
linear embedding. Place a mirror segment at each angle so that it makes
equal angles to each of the parts of the embedding that form the angle.
The two parts of the embedding at the angle now have the appearance of
rays being reflected by the mirror segment. Then the image of the folded
ribbon in the plane is by definition a bundle of parallel rays reflected by
the mirr or segments. See Figure 2 for an illustration of this geometry. It is
assumed that separate straight line bundles intersect either along mirrors,
or transversely in the pattern o f the self-crossings of the core diagram. A flat
ribbon immersion is such an immersion of a ribbon into the plane. A flat
knotted ribbon is a choice of weaving that overlies a flat ribbon immersion.
In such a weaving, the width of any straight interval of parallel rays must
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4 L. Kauffman
be chosen so that no bundle of parallel lines can overlap a fold line when
it is intended to weave that interval with the intervals meeting at the fold.
Such an overlap would be the same as a situation where the bundle of lines
overlapped one of the mirrors, and would have to be either entirely over, or
entirely under the mirror fold. Such a choice is obtained by first selecting an
over or under crossing at ea ch self-intersection of the center-line, and then
extending these choice s compatibly across the families of reflected rays. In
this way, any piecewise linear knot diagram gives rise to a collection of
flat knotted ribbons of varying width. For a given diagram, there will be a
unique larg est width knotted ribbon. We would like, within the topologica l
type of a given knot, to determine the least length of ribbon needed for a
given choice of width.
In order to discuss the issue of the length of a given knotted ribbon
(for a given width), there are two choices. We c an assume that the knotted
ribbon is in the traditional form of a closed piece w ise linear loop, and take
the leng th of the core curve. This is the closed length of the knotted ribbon.
Alternatively, we can choos e to cut and truncate the knotted ribbon and
measure the length of the unfolded ribbon that results from the cut. In some
cases, there is a clear way to make the truncation. For example, in Figure
3 we illustrate how a trefoil knot oc c urs in the form of a woven pentagon.
In this form it is natural to truncate the ribbons flush with the sides of the
pentagon. When there is such a choice, we shall call the resulting length
the truncation length of the knotted ribbo n.
In Figure 3, we illustrate the process of forming a trefoil knot and pulling
it to make a pentagonal flat knotted ribbon. We also show the internal
mirror structure of this kno t. The edges of a regular pentagon ar e the
mirror segments. A ray of light enters the pentagon and is r e flected three
times before exiting. In the diagram, the over and undercrossings of the
trefoil knot have been indicated. Note that the pentagonally folded trefoil
knot is seen here as a limiting case of flat banded knots whose mirrors are
segments o n the sides of a regular pentagon. There is no way, using this
particular geometr y, to extend the lengths of the mirro rs beyond the lengths
of the pentagonal sides a nd still maintain the possiblity of an embedding.
We also show in Figure 3 how a totally internally self-reflecting pentagon
would behave. In this case the alternating over and undercrossing pattern is
actually not realizable by str aight lines in three space, but could be effected
by a weaving process, as we have des crib e d.
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Minimal Flat Knotted Ribbons 5
Figure 3. A Pentagonal Mirror
3. The Trefoi l Knot, the Pentagon and the Golden Ratio
We begin with an e xperiment that one can perfo rm with the pattern of
the trefoil knot and a strip of pa per . Tie the strip into a trefoil and pull it
gently tight and fold it so that you obtain a flat k not. As you pull it tight,
a regular pentag on will appear.
It is intuitively clear that this pentagonal form of the trefoil knot uses
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6 L. Kauffman
the least length of pa per fo r a given width of paper (to make a flattened
trefoil). At this writing, we do not have a proof of this statement. It is the
purp ose of this se c tion to analyze the pentagonal geometry and to make a
quantitative version of this conjecture.
w
w
d
d d
l
l
l
l
w
w
(d-l)/2
Figure 4. Pull and Unfold Trefoil
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Minimal Flat Knotted Ribbons 7
We can determine the length to width ratio of the strip of paper obtained
from this pentagonal construction, by first cutting the knot exactly along
the edges of the pentagon, and then unfolding this strip. See Figure 4.
The lengths l and d that label the strip in Figure 4 are the edge length l
and the chord length d of the pentagon. To see this, contemplate Fig ure 4,
or better yet, perform the experiment of making the knot, cutting it flush
to the pentagon and unfolding the strip. The leng th of the whole strip is
L = 2(l + d).
In Figur e 4 there is a blown-up picture of the pentagon with the e dge-
length l, the chord d and the width of the strip W labeled. You can see
that the width is indeed the leng th o f a particular perpendicular dropped
to the chord. It is also apparent from this diagram that
W
2
+ (d − l)
2
/4 = l
2
.
Thus we can s olve for the width W of the strip in terms o f l and d. The
result is
W =
p
3l
2
− d
2
+ 2ld/2.
Since the length of the strip is L = 2(d + l), the ratio of length to width is
L/W = 4(d + l)/
p
3l
2
− d
2
+ 2ld
= 4((d/l) + 1)/
p
3 − (d/l)
2
+ 2(d/l).
Let
φ = d/l .
Then we have
L/W = 4(φ + 1)/
p
3 − φ
2
+ 2φ.
This is a formula for the leng th to width ratio of the flattened (trun-
cated) pentagonal trefoil knot. Now we use a clas sical fact about the pen-
tagon.
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8 L. Kauffman
Fact. T he ratio φ = d/l of the chord of a regular pentagon to its edge
length is equal to the go lden ratio. That is, φ = (1 +
√
5)/2 and φ
2
= φ + 1.
To see this fact, contemplate Figure 5.
l
d
d
(d+l)/d = d/l
d
Figure 5. The Pentagon
The small pentagon has its edge l and chord d labeled. We embed the
small pentag on in the larger one and observe via a parallelog ram and by
similar triangles that d/l = (d + l)/d. Thus, with φ = d/l, we have d/l =
1 + l/d whence φ = 1 + 1/φ. This is sufficient to show that φ is the golden
ratio. Note how the golden ratio appears here through the way that a
pentagon embeds in a pentagon, a pentagonal self-reference.
We conclude that
L/W = 4(φ + 1)/
p
3 − φ
2
+ 2φ = 4(φ + 1)/
p
2 + φ
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Minimal Flat Knotted Ribbons 9
Thus
L/W = 4(φ + 1)/
p
2 + φ = 4/
p
7 − 4φ.
(We thank Robert Gray
3
for observing this la st equality.)
This shows how the golden ratio appears crucially in this fundamental pa-
rameter associated with the trefoil knot.
Numerically, we find that
L/W = 5.5055276818846941528288383276435507 ···
We conjecture that
L/W == 4/
p
7 − 4φ = 4/
q
5 − 2
√
5
is indeed the minimum length to width ratio for a flatt en ed truncated trefoil
knot.
Remark. It is well-known in rec reational mathematics that the trefoil rib-
bon pr oduces a pentagon
6
. To our knowledge, this is the first time this
construction has been analyzed for its length to width ra tio, and it is the
first time that the construction has been placed in the context of the sear ch
for minimal ribbon length.
4. Figure Eight Knot and a Hexagon
Figure 6. Figure Eight Knot and its Hexagon
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10 L. Kauffman
In this section we consider the analogous question for the figure eight
knot. Figure 6 illustra tes the experiment for the figure eight knot that leads
to a flat folded hexagon. As the figure shows, the hexagon is not regular. It
has two opposite parallel sides of the same length, and four sides that are
equal in length to o ne another.
A
B
CD
E
F
G
H
0
A = (a,b)
B = (a+c,0) = ((4/3)a,0)
C = (a-c,2b) = ((2/3)a, 2b)
D = (0, 2b)
0 = (0, 0)
G = ((2/3)a, 0)
F = (a/2, b/2)
K
Figure 7. Geometry of Folde d Figure Eight Hexagon
Consider Figure 7, which shows the hexagon formed by the tightened
ribbon and the center line of the ribbon (appearing as five line segments
with four folds), together with a sixth line segment connecting the sta rting
point (A) and ending point (H) o f this folded center line. Place the figure
on a coordinate plane so that the point O = (0, 0), A = (a, b), and B =
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Minimal Flat Knotted Ribbons 11
(a + c, 0). Since A is the midpoint of BC, we must have C = (a −c, 2b) and
D = (0, 2b). Finally, since lines BC and DG are parallel,
G = B + (D − C) = ((a + c) − (a − c), (2b) − (2b)) = (2c, 0).
On the other hand, ∠AGB = ∠ABG because AG is pa rallel to DH, so
G = (a − c, 0) implying that a = 3c. Thus B = (
4
3
a, 0), C = (
2
3
a, 2b), and
G = (
2
3
a, 0).
Now the angle of incidence at A equals the angle of reflection, so ∠BAG =
∠CAF . Furthermore, F G and AB are parallel, so ∠BAG = ∠AGF and
∠CAF = ∠AF G. Hence ∠AF G = ∠AGF. Thus △GAF is isosceles, so
that |AF | = |AG|.
Since F is the intersectio n point of the lines OA and DG, it satisfies the
equations
y =
b
a
x and y =
−3b
a
x + 2b.
Hence F = (
a
2
,
b
2
). Then |AF |
2
= |AG|
2
implies
1
4
a
2
+
1
4
b
2
=
1
9
a
2
+ b
2
,
so that
b =
r
5
27
a.
The center line of the ribbon starts at (a, b) and proce e ds to
(0, −2b), (−a, b), (a, −b), and (0, 2b), be fore finishing at (−a, −b). Thus the
length o f the ribbon is
L = 4
p
a
2
+ 9b
2
+
p
4a
2
+ 4b
2
=
32
3
r
2
3
a
In order to find the width W of the ribbon, view Figure 8. Here we observe
that W/2 is the distance between the parallel lines through AB and F G.
It follows from our coordinate calculations that |AB| = |AG| since
|AB| = |(a, b) − (4a/3, 0)| = |((a, b) − (2a/3, 0)| = |AG|.
From this, and the equality of alternate interior angles between parallel
lines, it follows that the three angles indicated by α in Figure 9 are indeed
equal to one another. That is, ∠AGB = ∠ABG = ∠BGK . It then follows at
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12 L. Kauffman
once that W = (4/3)a sin(α) and that α = arctan(3b/a) = arctan(
p
5/3).
Hence
W = (4/ 3)a sin(arctan(
p
5/3)) = (4/3)a
p
5/3/
p
1 + 5/3 = a
√
10/3.
and so
L/W =
32
3
r
2
3
3
√
10
=
r
2
3
32
√
10
=
32
√
15
≈ 8.26236447190916 ··· .
G
0
W/2
B = ((4/3)a, 0)
G = ((2/3)a,0)
α
W = (4/3)a sin( )
α
A = (a, b)
αα
α
= arctan(3b/a)
K
F
Figure 8. Finding the Width W
As expected, the figure eight knot is longer than the trefoil knot. We con-
jecture that
32
√
15
is the minimal (truncation) length to width ratio for the
figure eight knot.
Even at the level of the figure eight knot there are other ways to make
the folds. But experiments so far reveal that the L/W ratio is larger for
these other constructions. An example of an alternative fold is shown in
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Minimal Flat Knotted Ribbons 13
Figure 9. We leave it to the reader to verify that this folding geometry
leads to more length than the semi-re gular hexagon described above.
Figure 9. An alternate, longer, flat figure eight knot
5. Discussion
Some workers have thoug ht about the relationship between thickness for
flat knots and thickness for knotted tube s. For example Kusner
4
informed
the author of this paper that one can get an upper bound of the form
Ropelength(K) ≤ c
1
Ribb onle ngth(K) + c
2
Cr ossing(K)
He remarks that it seems that the best one can expect for ribbo nlength is
a linear relation
c
3
Cr ossing(K) ≤ Ribbonlength(K) ≤ c
4
Cr ossing(K)
with minimum crossing number. Here Ropelength and Ribbonlength refer
respectively to minimal length to width ratios for tubular and flat knots,
while Crossing refers to the minimal crossing number.
It is clear that much more work remains to be done in this field. The
immediate appearance of geomerty, in relation to flattened knotted ribbons,
is encour aging. The problems of minimization discussed in this paper can
be e xplored both theoretically and via computer models. There should be
impo rtant and interesting relationships between the structure of ribbons ,
the structure of ropes, and geometry in three dimensiona l space. Values
such as the minimal rope length and minimal ribbonleng th for knots a re
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14 L. Kauffman
fundamental topological quanta associated with the knots. It is to be ex-
pected that such numbers will be related to the geometry, topology and
physics of these entities.
Acknowledgment. Most of this effort was sp onsored by the Defense Ad-
vanced Research Projects Agency (DARPA) and Air Force Research Labo-
ratory, Air Force Materiel Command, USAF, under agreement F30602-0 1-
2-05022. The U.S. Government is a uthorized to reproduce and distribute
reprints for Government purposes notwithstanding any copyright annota -
tions thereon. The vie ws and conclusions contained herein are those of the
authors and should not be interpr e ted as necessarily representing the of-
ficial policies or endorsements, either expressed o r implied, of the Defense
Advanced Research Projects Agency, the Air Force Research Laboratory,
or the U.S. Government. (Copyright 2004.) It gives the first author g reat
pleasure to acknowledge suppor t from NSF Grant DMS-024 5588, to give
thanks to the University of Waterloo and the Perimeter Institute in Water-
loo, Canada for their hospitality during the preparation of this research, and
to thank Jorge Alberto Calvo for penetrating remark s about the g e ometry
of the figure eight knot.
References
1. G. Buck and J. Simon, Thickness and crossing number of kn ots, Topology
and its Appl. 91 (1999), 245-257.
2. J. Cantarella, R . Kusner, J. Sullivan, On the minimum ropelength of knots
and links, 2001, arXiv:math.GT/0103224.
3. R. Gray, (private e-mail)
4. R. Kusner, (private e-mail)
5. E. Rawdon, Approximating the thickness of a knot. In A. Stasiak, V. Ka-
tritch, and L. Kauffman, editors, “Ideal Knots”, pp. 143-150. World Sci.
(1998).
6. M. Schneider, “A Beginner’s Guide to Constructing the Universe”, Harper
Collins Pub. (1994), p. 103, p. 230.
7. A. Stasiak, V . Katritch and L. Kauffman ( editors), “Ideal Knots”, World
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