KNOT CLASSIFICATION AND INVARIANCE
ELEANOR SHOSHANY ANDERSON
Abstract. A key concern of knot theory is knot equivalence; effective representation of these objects through various notation systems is another. This
paper will look at these areas’ intersection. It will examine several systems
used to describe knots, stressing three questions: (a) Do non-equivalent knots
have different designations under each system (i.e does each system classify
knots), (b) Do equivalent knots have the same designation under each system
(i.e. is the system a knot invariant), and (c) If the answer to (a) or (b) is
generally no, under what conditions does it become yes?
Contents
List of Figures
1. Mathematical Introduction to Knots
2. Planar Projections
3. Dowker Notation
4. Braid Notation
Acknowledgments
References
Figures
1
2
3
4
6
9
9
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List of Figures
1
Three knots
2
Wild knot
10
9
3
Unknot equivalent
10
4
Knot and projection
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5
Type I Reidemeister move
11
6
Type II Reidemeister move
11
7
Type III Reidemeister move
11
8
Tricolorability
12
9
Type II preserves tricolorability
12
10 Type III preserves tricolorability
13
11 Unknot equivalent
14
12 Generating Dowker notation
14
13 Diagramming from Dowker notation
15
14 Same Dowker notation
16
15 Composite knot
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ELEANOR SHOSHANY ANDERSON
16 Equivalent Dowker notation
17
17 Same Dowker notation
18
18 Equivalent knots
19
19 Braid and braid frame
20
20 Braid as projection
21
21 Labeling for braid representation
22
22 Ambient isotopy for braid representation
23
23 Eliminating superfluous arcs
24
24 Equivalent braids
24
25 Equivalent braids
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1. Mathematical Introduction to Knots
Definition 1.1. A knot is an embedding of the circle in 3-space.
By Definition , one of the easiest ways to construct a mathematical knot is tie
two ends of a string together. Figure 4 makes use of this to show three knots ranging from the simple knot called the unknot (a) to an apparently complex knot (c).
Under Definition 1, Figure 2, which shows a knot with infinitely many crossings
(the infinitude is represent by the ellipsis), due to R.H. Fox, is a knot. This is called
a wild knot because, informally speaking, it would take infinitely many uncrossings
to “undo” it. We wish to rule out this sort of behavior.
Definition 1.2. An ambient isotopy of a knot K is a continuous deformation of K
in R3 so that at any time the image of K is an embedded circle.
Definition 1.3. A knot is called tame if it is ambient isotopic to a simple closed
polygon in R3 .
Note that Definition 1.3 rules out Fox’s knot example because a polygon has
only finitely many sides.
Henceforward, this paper will consider only tame knots, and we say that two
knots are equivalent if one can be changed into the other by means of an ambient
isotopy. Figure 3 shows a knot equivalent to the unknot.
Because every tame knot is ambiently isotopic to a simple closed polygon, we
can speak of each tame knot K being defined by the set {p1 , . . . , pn }, where the
union of the line segments [p1 , p2 ], . . . , [pn−1 , pn ] and [pn , p1 ] form a closed polygon
isotopic to K. Note that this choice is not unique, as many polygons are isotopic
to each other.
Definition 1.4. The points p1 , . . . , pn , defined as above, are called the vertices of
K.
KNOT CLASSIFICATION AND INVARIANCE
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In the next three sections of this paper, we will examine three systems of knot
notation and see to what degree they classify (i.e. whether they distinguish nonequivalent knots) or act as invariants for knots (i.e. whether they assign the same
notation to equivalent knots).
2. Planar Projections
Planar projections, also called knot projections and knot diagrams, are extremely
important because they give visually accurate representations of knots. Figure 4
shows a knot (a) and its planar projection (b). The parts where it overlaps itself
are called overcrossings and undercrossings; the choice of term depends on our
perspective.
Definition 2.1. A projection is a map R3 → R2 .
Definition 2.2. Consider a knot K represented by a polygonal embedding and a
projection P : R3 → R2 . A point p ∈ P (K) is called a multiple point of P if P −1 (p)
contains more than one point and a double point of P if P −1 (p) contains exactly
two points.
Definition 2.3. A projection P : R3 → R2 is a planar projection of a knot K if
(1) There are only finitely many multiple points of P ;
(2) Each multiple point of P is a double point of P ;
(3) No vertices of K are mapped to double points;
(4) The over- and under-crossing lines of each double point are marked.
We will first show that planar projections classify knots. To do so, we introduce the three Reidemeister moves and their inverses, vital methods of diagram
manipulation.
Definition 2.4. A Reidemeister move is a transformation of of a planar projection.
The three types of move are shown in Figures 5, 6, and 7.
Definition 2.5. Two planar projections are called equivalent if one can be changed
into the other by a finite number of Reidemeister moves.
The next theorem shows why Reidemeister moves are so important, and why we
can claim that planar projections offer accurate physical representations of knots.
Theorem 2.6. Two knots are equivalent if and only if all of their planar projections
are equivalent.
The proof of this theorem can be found in [4]. While useful, it does not give any
upper bound on the number of Reidemeister moves needed to show equivalence; in
fact, no such bound is known.
We also have not shown that there exists more than one equivalence class of
knots. To do so, we introduce the idea of tricolorability, and apply the Reidemeister
moves to this concept.
Definition 2.7. A strand in a projection of knot is a segment of curve spanning
from under-crossing to under-crossing, with only over-crossings in between.
Definition 2.8. A knot projection is tricolorable if it meets the following criteria:
(1) Each of its strands can be colored in one of three colors;
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ELEANOR SHOSHANY ANDERSON
(2) Either one or three colors come together at each crossing;
(3) At least two colors are used.
The unknot in Figure 8(a) violates Item 3 of Definition 2.8. The knot in Figure
8(b) (called the trefoil knot) is tricolorable. As Figures 9 and 10 show, Type II and
III Reidemeister moves preserve tricolorability regardless of the strands’ coloring;
a Type I Reidemeister move does so because only one strand is involved. Since the
trefoil is tricolorable and the unknot is not, the unknot’s diagram is not equivalent
to the trefoil’s, and so by Theorem 2.6, the unknot is not equivalent to the trefoil.
This shows there are non-equivalent knots.
Although planar projections classify, they are only invariants of knots up to
equivalence by Reidemeister moves. Figure 11 shows a diagram of knot K which,
with a single Type I Reidemeister move, becomes the unknot’s diagram. Thus,
though K and the unknot have different diagrams, by Theorem 2.6 the knots are
equivalent.
3. Dowker Notation
For many applications, such as inputting the data of a knot into a computer,
pictures are insufficient. Dowker notation, which reduces any knot to a string of
integers, is useful for this; so useful, in fact, that in 1998 Morwen Thistlethwaite
used it to list all knots of fewer than fourteen crossings [5].
Here is how to generate Dowker notation from a knot. Take a projection and
choose any crossing, and label the under-strand with a 1. Pick a direction in which
to follow that strand. At the next crossing, label that strand with a 2. Continue
following the same strand in the same direction, labeling the strands consecutively,
until arriving back at 1. The result should look like Figure 12.
Notice that each of this figure’s crossings is denoted by one odd and one even
integer. In fact, this will always be the case.
Theorem 3.1. Each crossing of any knot labeled in the Dowker style will always
be denoted by one even and one odd integer.
Proof. Consider a loop L of a knot diagram created by a crossing labeled by two
odd numbers or two even numbers. Because the overstrand and understrand are
of the same parity, this means that strictly between them is an odd number of
crossings between the two along L. (For example, if the crossing creating L is
labeled 2 6, then the crossings labeled 3, 4, and 5 are between 2 and 6.) This,
however, is impossible. Since L is a closed curve, each segment that enters L must
also leave it, producing an even number of crossings. Thus no such loop L can exist;
consequently all crossings must be labeled with one even and one odd integer. We can thus pair the even and odd integers and list them like so, using Figure
12 as an example and + and − to show whether each strand goes over or under:
−1
+6
+3
−14
−5 +7 −9 +11
+16 −12 +2 −4
+13 −15
−8 +10
KNOT CLASSIFICATION AND INVARIANCE
5
Since each pair has one + and one − (as every crossing consists of one overand one under-stand), and since the top row goes in a predictable order, we can
eliminate it. Thus in Dowker notation Figure 12 is called
+6 −14 +16 −12 +2 −4 −8 +10.
It is also possible to (re)construct a knot from Dowker notation. As an illustration, we do this with the following example:
+4 +8 +10 +2 +6.
First, we know that each integer in the notation must be matched to its even
partner, and that partner must have the opposite sign. Thus this Dowker notation
is shorthand for
−1 −3 −5 −7 −9
+4 +8 +10 +2 +6.
Begin by drawing the first crossing, labeling it 1 and 4, like Figure 13(a); make
sure the 1 strand goes under the 4 strand. Continue on, drawing crossings and labeling them accordingly. When an integer is reached which has already been used to
label a crossing, circle back and pass through that crossing on the already-labeled
strand, then continue creating crossings as needed; be careful not to create extra crossings. Figure 13(b) shows this process in action. Finally, when all crossings
have been made, loop the two ends together. Figure 13(c) shows the complete knot.
One problem with Dowker notation is that non-equivalent knots can have the
same Dowker notation. Two examples of this problem, are discussed below. To
understand the first, here are two definitions.
Definition 3.2. The mirror image of a knot K is the knot obtained by changing
K’s over-crossings to under-crossings and vice versa.
Definition 3.3. A knot is amphicheiral if it is equivalent to its mirror image.
One problem, as shown in Figure 14, is that there are two ways to loop back to
an already-used integer: these loops can go either up or down, resulting in a knot
and its mirror image. In fact, these two knots are the trefoil knot and its mirror
image, which Max Dehn [3] showed are not equivalent.
A second problem with Dowker notation arises from mirror-image complications
with composite knots.
Definition 3.4. Suppose that K1 and K2 are knot embeddings meeting a plane E
only along the arc from p to q. Remove this arc from both knots to and join them
together at p and q to form a new knot K, which is called the composition of K1
and K2 and denoted K = K1 #K2 . Figure 15 illustrates this.
Definition 3.5. A knot is called composite if it is the composition of two knots,
neither of which are the unknot. A knot K is called prime if the only decomposition
K = K1 #K2 of it has K1 or K2 as the unknot.
When Dowker notation strings include subpermutations of the even numbers
(e.g. +4 +6 +2 +10 +12 +8, where +4 +6 +2 is a subpermutation of 2 4 6 and
+10 +12 +8 is a subpermutation of 8 10 12), the result is a composite knot. This
is because the first subpermutation forms one knot, complete but for the closing,
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ELEANOR SHOSHANY ANDERSON
and the latter subpermutations produce others.
Figure 16 shows that +4 +6 +2 completely describes a knot (call it Knot A) in
Dowker notation, and, if we label our crossing beginning with -7, then +10 +12
+8 completely describes another knot (Knot B) in Dowker notation. Figure 16
shows these knots with their ends unclosed, to show how A might attach to B. As
discussed above, both strings of notation can result in a knot and its mirror image.
When composing the two strings together to make the knot +4 +6 +2 +10 +12
+8, therefore, we have four choices:
(1) We can join Knot A to Knot B;
(2) We can join Knot A to Knot B’s mirror image;
(3) We can join Knot A’s mirror image to Knot B;
(4) We can join Knot A’s mirror image to Knot B’s mirror image.
The results of option 1 and option 2 are given in Figure 17; Adams [1] shows
these are not equivalent knots.
These problems outline the restrictions necessary for Dowker notation to be used
for classification. If knots described by Dowker notation including subpermutations–
i.e. composite knots–are excluded, a particular string of notation may describe only
a knot and its mirror image [1]. So, restricting ourselves to prime amphicheiral knots
allows Dowker notation to be used for classification.
With exactly these restrictions, Thistlethwaite was able to make great use of
Dowker notation. However, it is not as useful as an invariant. Figure 18 shows
diagrams of equivalent knots (to get from the left one to the right, simply perform
one Type I Reidemeister move). The top diagram is denoted +2; the bottom is
denoted +2 +4. Thus, while Dowker notation–with several restrictions–is a very
useful classifier, it is useless as an invariant.
4. Braid Notation
A particularly beautiful way of representing knots is by using braid notation.
First, we describe braids and their properties; then we examine braid notation.
Definition 4.1. Let ABCD be a rectangle in R3 with points P1 , . . . , Pn on side
A and Q1 , . . . , Qn arranged on side C. For each i with 1 ≤ i < n, |Pi+1 − Pi | =
|Qi+1 − Qi |, and P1 and Q1 are equal distance from side B.
Let si , 1 ≤ i ≤ n, be simple disjoint arcs from Pi to Qπ(i) , where i → π(i) is a
bijective function from {1, . . . , n} to itself. This function must also not have any
subpermutations of {1, . . . , n} in the sequence {π n (1)}, because these would result
in multiple disjoint knots, collectively called a link.
Each si meets any plane perpendicular to sides B and D exactly once. When we
identify each Pi with its respective Qi , we call the set {si } a braid. Rectangle
ABCD is called the braid frame.
Example 4.2. Figure 19 shows a representation of a braid with its braid frame.
Keep in mind that each Pi is identified with its respective Qi , and that in a twodimensional representation, we must indicate which strand crosses which.
Figure 20 shows another drawing of a braid, with the long strings around the
sides showing the identification of the top and bottom of the braid frame. We will
KNOT CLASSIFICATION AND INVARIANCE
7
continue to depict braids as in Figure 19, because it is more convenient to leave out
the long looping strings, but this picture shows that this braid representation is a
valid knot projection, and so by Theorem 2.6, it corresponds to a knot.
This shows that at least the knot corresponding to Figure 20 can be represented
as a braid. Our next goal is to show that in fact all knots can so represented. First,
three helpful definitions are given below.
Definition 4.3. A braid B is closed if there is a choice of axis A perpendicular to
the plane formed by the braid frame such that, starting from point S on B, B can
be traversed clockwise about A without backtracking, and arriving back at S.
Definition 4.4. A maximal overpass of a knot diagram is a subarc of the diagram
going over at least one crossing and terminating on each end at undercrossings,
but not including the undercrossings. A minimal underpass of a knot diagram is
a subarc of the diagram going from overcrossing to overcrossing, without including
the overcrossings.
Note that by the definition of maximal overpass, as we traverse any projection
of a knot, we must alternate maximal overpasses and minimal underpasses. Thus
the number of maximal overpasses equals the number of minimal underpasses.
Definition 4.5. The bridge number of a knot is the least number of maximal
overpasses found in any projection of the knot.
Theorem 4.6. Any knot can be represented as a closed braid.
Proof. Consider some projection P of some knot K. Suppose P has m maximal
overpasses. Choosing two points on each maximal overpass gives the 2m points
R1 , . . . , R2m , with, for 1 ≤ i ≤ m, arc si connecting R2i−1 and R2i , and arc ti
connecting R2i and R2i+1 . Note that the crossings made by each segment si are
overpasses and those made by each ti are underpasses, and that there are m si and
m ti . An example of this labeling is given by Figure 21.
By ambient isotopy we can arrange the ti to form m straight parallel segments
bisected by segment A; this rearrangement is shown by Figure 22. Now, each si
meets A an odd number of times (since each starts at R2i−1 and ends at R2i , each
si must begin on one side of A and end on the other); call these meeting points
Ri1 , Ri2 , . . . . As things stand, the si may cross a more than once. Thus we take
all subarcs of each si above A aside from the one connected to R2i−1 and “push”
them below A by ambient isotopy, as shown in Figure 23. Now each si and each ti
meets A at exactly one point.
Now, we can eliminate the ti from our representation, instead identifying R2i
with R2i+1 for 1 ≤ i ≤ m, since these are the points connected by ti . The result is
exactly a braid as defined by Definition 4.3.
Notation for a braid on n strands is given by the symbols σ11 , . . . , σn1 and σ1−1 , . . . , σn−1 ..
The symbol σi+1 means that the ith strand (counting from the left) crosses in front
of the (i + 1)th strand; the symbol σi−1 means that the ith strand crosses behind
the (i + 1)th strand. The crossings are read from top to bottom, and the sequence
of sigmas describing this reading are called the braid’s word. For example, Figure
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ELEANOR SHOSHANY ANDERSON
19’s word is σ1 σ2 .
Because of Theorem 4, we see that braid notation does classify knots. If two
braids have the same word, their braid representations will be ambiently isotopic to
each other. Since, as noted above, these braid representations are knot diagrams,
and since knot diagrams classify knots, we see that braid notation classifies knots.
However, braid words are not knot invariants. Figure 24 shows two braids between which an isotopy clearly exists–informally, one must simply “pull up” the
first crossing and “push down” the second crossing in (a) to attain (b)–making
them equivalent knot diagrams, and hence equivalent knots.
However, just like Reidemeister moves for projections, there are three rules for
word equivalence; these are called the equivalence moves, and in fact the first two
derive from the Reidemeister moves. First, one Type II Reidemeister move removes
any part of a word with σi σi−1 or σi−1 σi , since these simply denote a strand crossing
over or under its neighbor twice in succession.
Figure 25 describes the second equivalence move: it shows, by applying a Type
III Reidemeister move that segments of words denoted by σi σi+1 σi are equivalent
to the segment σi+1 σi σi+1 . Note that the strands’ initial and final positions are
unchanged, which is why this move is acceptable.
The third equivalence move comes not from the Reidemeister moves, but makes
use of what Figure 24 shows: if the ith and jth strands from the left are separated
by at least one strand, and the braid’s word contains the segment σ i σ j , then that
segment can be replaced by σ j σ i . As noted above, this is simply an ambient isotopy.
These three equivalence moves completely describe braid-word equivalence:
Theorem 4.7. Two braids’ words represent the same knot if and only if we can
get from one word to the other by a sequence of the above three equivalence moves.
Proof. First, suppose words w and x represent projections P and Q of knot K,
respectively. By Theorem 2.6, we can thus get from P to Q by a series of Reidemeister moves. If Type I moves are used, this will not affect w, as a braid word
only notes when separate strands cross. If Type II moves are used, they alter the
w by the first equivalence move, which is acceptable. If Type II moves are used,
w will be altered by the second equivalence move, which is also acceptable. Also,
any ambient isotopies performed during the Reidemeister sequence are acceptable,
because this is the third equivalence move.
To show the converse, suppose braid words w and x are linked by a sequence of
equivalence moves. Since each equivalence move is composed only of Reidemeister
moves and ambient isotopies, the knots represented by w and x are linked by a
sequence of Reidemeister moves and ambient isotopies, and so by Theorem 2.6, the
two knots are equivalent.
KNOT CLASSIFICATION AND INVARIANCE
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Acknowledgments
It is a pleasure to thank my mentors, William Lopes and Katie Mann, for
their enthusiasm, guidance, and great knowledge. Thanks also to my mother, Gila
Shoshany, for her great help in producing such elegant figures.
References
[1] Colin Adams. The Knot Book. American Mathematical Society. 2001.
[2] Gerhard Burde and Heiner Zieschang. Knots, Second Edition. De Gruyter Studies in Mathematics. 2003.
[3] Max Dehn. Die beiden Kleeblattschlingen. Mathematische Annalen Volume 75, Number 3,
1914.
[4] Lou Kauffman. Knot Diagrammatics. In Handbook of Knot Theory, ed. W. Menasco and M.
Thistlethwaite, 2005.
[5] Morwen Thistlethwaite et al. The First 1,701,396 Knots. Math Intelligencer Volume 20, Number 4. 1998.
Figures
Figure 1. Three knots
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ELEANOR SHOSHANY ANDERSON
Figure 2. Wild knot
Figure 3. Unknot equivalent
Figure 4. Knot and projection
KNOT CLASSIFICATION AND INVARIANCE
Figure 5. Type I Reidemeister move
Figure 6. Type II Reidemeister move
Figure 7. Type III Reidemeister move
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Figure 8. Tricolorability
Figure 9. Type II preserves tricolorability
KNOT CLASSIFICATION AND INVARIANCE
Figure 10. Type III preserves tricolorability
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Figure 11. Unknot equivalent
Figure 12. Generating Dowker notation
KNOT CLASSIFICATION AND INVARIANCE
Figure 13. Diagramming from Dowker notation
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Figure 14. Same Dowker notation
KNOT CLASSIFICATION AND INVARIANCE
Figure 15. Composite knot
Figure 16. Equivalent Dowker notation
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Figure 17. Same Dowker notation
KNOT CLASSIFICATION AND INVARIANCE
Figure 18. Equivalent knots
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Figure 19. Braid and braid frame
KNOT CLASSIFICATION AND INVARIANCE
Figure 20. Braid as projection
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Figure 21. Labeling for braid representation
KNOT CLASSIFICATION AND INVARIANCE
Figure 22. Ambient isotopy for braid representation
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Figure 23. Eliminating superfluous arcs
Figure 24. Equivalent braids
Figure 25. Equivalent braids