KNOT CLASSIFICATION AND INVARIANCE

KNOT CLASSIFICATION AND INVARIANCE

ELEANOR SHOSHANY ANDERSON

Abstract. A key concern of knot theory is knot equivalence; effective repre-

sentation of these objects through various notation systems is another. This

paper will look at these areas’ intersection. It will examine several systemsused 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 11. Mathematical Introduction to Knots 22. Planar Projections 33. Dowker Notation 44. Braid Notation 6Acknowledgments 9References 9Figures 9

List of Figures

1 Three knots 9

2 Wild knot 10

3 Unknot equivalent 10

4 Knot and projection 10

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 171

2 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 24

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 tietwo ends of a string together. Figure 4 makes use of this to show three knots rang-ing 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 calleda wild knot because, informally speaking, it would take infinitely many uncrossingsto “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 Kin 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 closedpolygon in R3.

Note that Definition 1.3 rules out Fox’s knot example because a polygon hasonly finitely many sides.

Henceforward, this paper will consider only tame knots, and we say that twoknots are equivalent if one can be changed into the other by means of an ambientisotopy. Figure 3 shows a knot equivalent to the unknot.

Because every tame knot is ambiently isotopic to a simple closed polygon, wecan speak of each tame knot K being defined by the set {p1, . . . , pn}, where theunion of the line segments [p1, p2], . . . , [pn−1, pn] and [pn, p1] form a closed polygonisotopic to K. Note that this choice is not unique, as many polygons are isotopicto each other.

Definition 1.4. The points p1, . . . , pn, defined as above, are called the vertices ofK.

KNOT CLASSIFICATION AND INVARIANCE 3

In the next three sections of this paper, we will examine three systems of knotnotation and see to what degree they classify (i.e. whether they distinguish non-equivalent knots) or act as invariants for knots (i.e. whether they assign the samenotation to equivalent knots).

2. Planar Projections

Planar projections, also called knot projections and knot diagrams, are extremelyimportant because they give visually accurate representations of knots. Figure 4shows a knot (a) and its planar projection (b). The parts where it overlaps itselfare called overcrossings and undercrossings; the choice of term depends on ourperspective.

Definition 2.1. A projection is a map R3 → R2.

Definition 2.2. Consider a knot K represented by a polygonal embedding and aprojection 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 exactlytwo 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 intro-duce the three Reidemeister moves and their inverses, vital methods of diagrammanipulation.

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 changedinto the other by a finite number of Reidemeister moves.

The next theorem shows why Reidemeister moves are so important, and why wecan 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 projectionsare equivalent.

The proof of this theorem can be found in [4]. While useful, it does not give anyupper bound on the number of Reidemeister moves needed to show equivalence; infact, no such bound is known.

We also have not shown that there exists more than one equivalence class ofknots. To do so, we introduce the idea of tricolorability, and apply the Reidemeistermoves to this concept.

Definition 2.7. A strand in a projection of knot is a segment of curve spanningfrom 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;


(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 Figure8(b) (called the trefoil knot) is tricolorable. As Figures 9 and 10 show, Type II andIII Reidemeister moves preserve tricolorability regardless of the strands’ coloring;a Type I Reidemeister move does so because only one strand is involved. Since thetrefoil is tricolorable and the unknot is not, the unknot’s diagram is not equivalentto 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 toequivalence 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 areequivalent.

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 ofintegers, is useful for this; so useful, in fact, that in 1998 Morwen Thistlethwaiteused it to list all knots of fewer than fourteen crossings [5].

Here is how to generate Dowker notation from a knot. Take a projection andchoose any crossing, and label the under-strand with a 1. Pick a direction in whichto follow that strand. At the next crossing, label that strand with a 2. Continuefollowing 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 eveninteger. In fact, this will always be the case.

Theorem 3.1. Each crossing of any knot labeled in the Dowker style will alwaysbe denoted by one even and one odd integer.

Proof. Consider a loop L of a knot diagram created by a crossing labeled by twoodd numbers or two even numbers. Because the overstrand and understrand areof the same parity, this means that strictly between them is an odd number ofcrossings between the two along L. (For example, if the crossing creating L islabeled 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 mustalso 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 Figure12 as an example and + and − to show whether each strand goes over or under:

−1 +3 −5 +7 −9 +11 +13 −15+6 −14 +16 −12 +2 −4 −8 +10


Since each pair has one + and one − (as every crossing consists of one over-and one under-stand), and since the top row goes in a predictable order, we caneliminate 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 illustra-

tion, 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 evenpartner, and that partner must have the opposite sign. Thus this Dowker notationis 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); makesure the 1 strand goes under the 4 strand. Continue on, drawing crossings and la-beling them accordingly. When an integer is reached which has already been used tolabel a crossing, circle back and pass through that crossing on the already-labeledstrand, then continue creating crossings as needed; be careful not to create ex-tra crossings. Figure 13(b) shows this process in action. Finally, when all crossingshave 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 thesame Dowker notation. Two examples of this problem, are discussed below. Tounderstand the first, here are two definitions.

Definition 3.2. The mirror image of a knot K is the knot obtained by changingK’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 toan already-used integer: these loops can go either up or down, resulting in a knotand its mirror image. In fact, these two knots are the trefoil knot and its mirrorimage, which Max Dehn [3] showed are not equivalent.

A second problem with Dowker notation arises from mirror-image complicationswith composite knots.

Definition 3.4. Suppose that K1 and K2 are knot embeddings meeting a plane Eonly along the arc from p to q. Remove this arc from both knots to and join themtogether 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 decompositionK = 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. Thisis because the first subpermutation forms one knot, complete but for the closing,


and the latter subpermutations produce others.

Figure 16 shows that +4 +6 +2 completely describes a knot (call it Knot A) inDowker notation, and, if we label our crossing beginning with -7, then +10 +12+8 completely describes another knot (Knot B) in Dowker notation. Figure 16shows these knots with their ends unclosed, to show how A might attach to B. Asdiscussed 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] showsthese are not equivalent knots.

These problems outline the restrictions necessary for Dowker notation to be usedfor classification. If knots described by Dowker notation including subpermutations–i.e. composite knots–are excluded, a particular string of notation may describe onlya knot and its mirror image [1]. So, restricting ourselves to prime amphicheiral knotsallows Dowker notation to be used for classification.

With exactly these restrictions, Thistlethwaite was able to make great use ofDowker notation. However, it is not as useful as an invariant. Figure 18 showsdiagrams of equivalent knots (to get from the left one to the right, simply performone Type I Reidemeister move). The top diagram is denoted +2; the bottom isdenoted +2 +4. Thus, while Dowker notation–with several restrictions–is a veryuseful 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 sideA 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 abijective function from {1, . . . , n} to itself. This function must also not have anysubpermutations of {1, . . . , n} in the sequence {πn(1)}, because these would resultin multiple disjoint knots, collectively called a link.Each si meets any plane perpendicular to sides B and D exactly once. When weidentify each Pi with its respective Qi, we call the set {si} a braid. RectangleABCD 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 two-dimensional representation, we must indicate which strand crosses which.

Figure 20 shows another drawing of a braid, with the long strings around thesides showing the identification of the top and bottom of the braid frame. We will


continue to depict braids as in Figure 19, because it is more convenient to leave outthe long looping strings, but this picture shows that this braid representation is avalid 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 representedas 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 tothe plane formed by the braid frame such that, starting from point S on B, B canbe 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 diagramgoing over at least one crossing and terminating on each end at undercrossings,but not including the undercrossings. A minimal underpass of a knot diagram isa subarc of the diagram going from overcrossing to overcrossing, without includingthe overcrossings.

Note that by the definition of maximal overpass, as we traverse any projectionof a knot, we must alternate maximal overpasses and minimal underpasses. Thusthe number of maximal overpasses equals the number of minimal underpasses.

Definition 4.5. The bridge number of a knot is the least number of maximaloverpasses 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 maximaloverpasses. Choosing two points on each maximal overpass gives the 2m pointsR1, . . . , R2m, with, for 1 ≤ i ≤ m, arc si connecting R2i−1 and R2i, and arc ticonnecting R2i and R2i+1. Note that the crossings made by each segment si areoverpasses and those made by each ti are underpasses, and that there are m si andm ti. An example of this labeling is given by Figure 21.

By ambient isotopy we can arrange the ti to form m straight parallel segmentsbisected by segment A; this rearrangement is shown by Figure 22. Now, each simeets A an odd number of times (since each starts at R2i−1 and ends at R2i, eachsi must begin on one side of A and end on the other); call these meeting pointsRi1 , Ri2 , . . . . As things stand, the si may cross a more than once. Thus we takeall 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 timeets 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 isexactly a braid as defined by Definition 4.3. �

Notation for a braid on n strands is given by the symbols σ11 , . . . , σ

1n and σ−1

1 , . . . , σ−1n ..

The symbol σ+1i means that the ith strand (counting from the left) crosses in front

of the (i + 1)th strand; the symbol σ−1i means that the ith strand crosses behind

the (i+ 1)th strand. The crossings are read from top to bottom, and the sequenceof sigmas describing this reading are called the braid’s word. For example, Figure


19’s word is σ1σ2.

Because of Theorem 4, we see that braid notation does classify knots. If twobraids have the same word, their braid representations will be ambiently isotopic toeach 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 be-tween which an isotopy clearly exists–informally, one must simply “pull up” thefirst crossing and “push down” the second crossing in (a) to attain (b)–makingthem equivalent knot diagrams, and hence equivalent knots.

However, just like Reidemeister moves for projections, there are three rules forword equivalence; these are called the equivalence moves, and in fact the first twoderive from the Reidemeister moves. First, one Type II Reidemeister move removesany part of a word with σiσ−1

i or σ−1i 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 TypeIII Reidemeister move that segments of words denoted by σiσi+1σi are equivalentto the segment σi+1σiσi+1. Note that the strands’ initial and final positions areunchanged, which is why this move is acceptable.

The third equivalence move comes not from the Reidemeister moves, but makesuse of what Figure 24 shows: if the ith and jth strands from the left are separatedby at least one strand, and the braid’s word contains the segment σiσj , then thatsegment can be replaced by σjσi. 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 canget 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 Reide-meister moves. If Type I moves are used, this will not affect w, as a braid wordonly notes when separate strands cross. If Type II moves are used, they alter thew 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 ofequivalence moves. Since each equivalence move is composed only of Reidemeistermoves and ambient isotopies, the knots represented by w and x are linked by asequence of Reidemeister moves and ambient isotopies, and so by Theorem 2.6, thetwo knots are equivalent. �


Acknowledgments

It is a pleasure to thank my mentors, William Lopes and Katie Mann, fortheir enthusiasm, guidance, and great knowledge. Thanks also to my mother, GilaShoshany, 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 Mathe-

matics. 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, Num-

ber 4. 1998.

Figures

Figure 1. Three knots


Figure 2. Wild knot

Figure 3. Unknot equivalent

Figure 4. Knot and projection


Figure 5. Type I Reidemeister move

Figure 6. Type II Reidemeister move

Figure 7. Type III Reidemeister move


Figure 8. Tricolorability

Figure 9. Type II preserves tricolorability


Figure 10. Type III preserves tricolorability


Figure 11. Unknot equivalent

Figure 12. Generating Dowker notation


Figure 13. Diagramming from Dowker notation


Figure 14. Same Dowker notation


Figure 15. Composite knot

Figure 16. Equivalent Dowker notation


Figure 17. Same Dowker notation


Figure 18. Equivalent knots


Figure 19. Braid and braid frame


Figure 20. Braid as projection


Figure 21. Labeling for braid representation


Figure 22. Ambient isotopy for braid representation


Figure 23. Eliminating superfluous arcs

Figure 24. Equivalent braids

Figure 25. Equivalent braids

KNOT CLASSIFICATION AND INVARIANCE

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