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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

Sci. Pub. (1998), Vol 19 Series on Knots and Everything, (414 pages).