The opening chapter offers activities that explore the world of knots and links — including games with knots — and invites the reader to generate their own questions in knot theory. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Additional topics include combinatorial knot invariants, knot polynomials, unknotting operations, and virtual knots.
The opening chapter offers activities that explore the world of knots and links — including games with knots — and invites the reader to generate their own questions in knot theory. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Additional topics include combinatorial knot invariants, knot polynomials, unknotting operations, and virtual knots.
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Overview
The opening chapter offers activities that explore the world of knots and links — including games with knots — and invites the reader to generate their own questions in knot theory. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Additional topics include combinatorial knot invariants, knot polynomials, unknotting operations, and virtual knots.
Product Details
| ISBN-13: | 9780486804637 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 01/18/2017 |
| Series: | Aurora: Dover Modern Math Originals |
| Pages: | 192 |
| Sales rank: | 605,094 |
| Product dimensions: | 5.90(w) x 9.00(h) x 0.50(d) |
About the Author
Inga Johnson is Professor of Mathematics at Willamette University.
Read an Excerpt
An Interactive Introduction to Knot Theory
By Inga Johnson, Allison Henrich
Dover Publications, Inc.
Copyright © 2017 Inga Johnson and Allison HenrichAll rights reserved.
ISBN: 978-0-486-80463-7
CHAPTER 1
Playing & Building Intuition
Each section in Chapter 1 is a hands-on introduction to knots, links, and equivalence. In this chapter, we will learn about several foundational concepts in the study of knots. The formal definition of a knot is postponed to Chapter 2 to allow time to first play and build intuition within the mathematically rich and beautiful field of knot theory.
Informally, a knot is a closed loop in space. The term closed loop means that the loop has no loose ends, and no beginning or ending points. You can think of a knot as a knotted-up circle made of string or wire. A link is a collection of closed loops in space and the number of loops is called the number of components of the link. A link can have one component. Thus, knots are just special types of links having only one component. Note that, when we use the term 'link' in this book, we are generally referring to both knots and links with more than one component.
The examples of knots and links seen in Table 1.1 are flat drawings of 3-dimensional loops in space. A 2-dimensional drawing of a link is called a diagram of the link. In a diagram, the term crossing is used to describe a location where one portion of the link passes over another portion of the link. Crossings are identified by a short break in the drawing of the curve, which indicates that this portion of the curve is passing under the unbroken portion of curve.
Two links are called equivalent if they have the same number of components and they can be physically manipulated in space (rotated, bent, twisted, stretched, etc.), without cutting, so that the first link is transformed identically into the second. We imagine that our strings are highly elastic so they can be scaled up or down in size, stretched and contracted.
While playing and building your intuition with the activities in this chapter, you may come up with your own questions or conjectures. We encourage you to collect and write down your ideas and add them to the list of questions in Section 1.8. Perhaps you will create your own new open research question about knots or perhaps you will stumble upon the same questions that the founders of the field of knot theory have puzzled over for years.
One last note before you begin. In mathematics, formal proofs of theorems rely on concepts and constructions being formally defined. Since the formal definition of a knot is not given in Chapter 1, we will not be asking the reader for formal proofs in this chapter. Instead, we ask the reader to give an argument that a statement is mathematically valid. We specifically use the term 'argument' rather than 'proof to allow for what we recognize is a tension between the desire for a formal proof with an initial lack of the formal definitions that are needed to make such a proof rigorous. An argument, in this setting, may be viewed as being less formal than a proof, but it should be as clear and complete an explanation as possible. In Chapter 2, you will notice a shift from play to formalism. Formal proofs related to the definition of a knot or link will serve as the foundation both for our play and for our proofs in the remainder of the book.
1.1 Projections, Diagrams & Equivalence
Exercise 1.1.1. (a) Identify the knots in Table 1.1. Build them with pipe cleaners, and determine any equivalences between different pictures. Record your findings and conjectures. (b) Identify the links with more than one component in Table 1.1. Build them with pipe cleaners, and determine any equivalences between different pictures. Record your findings and conjectures.
Given a link L in space and a light source some distance away, the shadow of the link made on a plane across from the light source is called a projection or shadow of the link. Projections can look similar to the pictures in Table 1.1, but they are missing information about which is the under-strand and which is the over-strand. The curve intersections in a projection are called precrossings. A precrossing is said to have been resolved once we have selected the crossing information (that is, we have specified which strand passes over and which passes under at the crossing). Once all crossing information is determined in a link projection, the image is then a link diagram.
The projection of a link L onto two distinct planes in can result in strikingly different images. Some projections are nonstandard and cannot be used to recreate the link in space, even if crossing information is included. While such projections may be useful in certain situations, they are not the projections that knot theorists typically study. (A list of specific projections characteristics to avoid will be studied in Chapter 2.) Two different projections of a link, L, can also have vastly different numbers of crossings. The following two exercises investigate the relationship between a link, its numerous projections, and the diagrams stemming from those projections.
Exercise 1.1.2. (a) Make a knot out of rigid material such as wire. Draw a projection of that knot from two perspectives that result in significantly different projections. (b) For that same knot, draw a nonstandard diagram of your knot from which the knot cannot be reconstructed. (For instance, maybe there are places in your diagram where several strands are tangent to each other or where three or more strands of the knot intersect at a single point.)
Exercise 1.1.3. Create a single rigid knot K in space such that one projection of K results in a diagram with four crossings and another projection results in a diagram with zero crossings.
Exercise 1.1.4. The knot labeled (a) in Table 1.1 is called the unknot or the trivial knot. Give an argument explaining why any knot diagram with exactly one or exactly two crossings must be equivalent to the unknot. (Hint: Draw the crossing(s) first, and then make a knot by connecting the ends in all possible ways so that no more crossings are created. Approach this task systematically so that your argument shows, without a doubt, that all possible ways have been considered.)
We've just seen that there are no nontrivial knots that can be drawn with just one or two crossings. In fact, the smallest nontrivial knot is a knot that can be drawn with three crossings. Any knot that can be drawn with three crossings and no fewer is typically referred to as a trefoil knot.
1.2 Crossing and Unknotting Numbers
Some of the links in Table 1.1 can be manipulated in space and then redrawn with fewer crossings. The crossing number of a link L is the minimum number of crossings needed in a diagram of L. For instance, if the crossing number of L is five, then it is impossible to draw a diagram of L that has four or fewer crossings.
Exercise 1.2.1. For each link in Table 1.1, make a conjecture about the crossing number of the link. Can you provide an argument in support of any of your conjectures? Which conjectures do you not yet have enough tools to prove?
Exercise 1.2.2. Determine the crossing number of the knot in Figure 1.2.1.
Exercise 1.2.3. Consider the knot projection in Figure 1.2.2. Start with an arbitrary precrossing of P and resolve it into a crossing. Imagine grasping the over-strand and pulling this strand up out of the plane of the paper. Using this visualization as inspiration, show for your projection that the remaining precrossings can be resolved so that the resulting knot is the unknot.
Exercise 1.2.4. Use the example in Exercise 1.2.3 as a guide to write an argument that given any knot projection, the precrossings can be resolved to produce a diagram of the unknot. (Hint: There are two parts to this problem. First you must devise an algorithm to create the desired unknot diagram. Then you must argue that the resulting diagram actually is the diagram of the unknot.)
Let K be a knot. Using your argument in Exercise 1.2.4, we can show that, given any diagram D for K, some number of crossings can be changed in D to produce a diagram of the unknot. Note that changing a crossing in a knot diagram is like passing the corresponding knot through itself in space. Typically, this "passing through" operation results in a different type of knot. The minimum number of times the knot must pass through itself before it becomes unknotted is called the unknotting number of the knot.
Note that the definition of unknotting number is a spatial characteristic of the physical knot. Thus, the unknotting number is not tied to a particular diagram of the knot. We can use diagrams to build intuition about what a knot's unknotting number might be, but such intuition does not always constitute a proof that a given knot has a specific unknotting number. For instance, if we can show that changing two crossings in a diagram of a knot K produces the unknot, we can say that the unknotting number of K is at most two. We cannot, however, claim that the unknotting number of K is exactly two without more work. Perhaps there's another diagram of the knot that only requires one crossing change to become unknotted.
Exercise 1.2.5. Use the diagrams provided to conjecture the unknotting number of the knots labeled (b), (f), and (o) from Table 1.1. Experiment with alternate diagrams for these knots to give additional support for your conjectures.
We will take a closer look at the unknotting number and study interesting ways to unknot diagrams in Chapter 7.
1.3 Alternating Knots
A diagram D for a link L is called an alternating diagram provided that, if you traverse each component in the diagram, then you alternately pass over and under the crossings. For example, the diagram (b) from Table 1.1 is alternating, while the diagram (1) is not. A link L is called an alternating link provided that there exists an alternating diagram D for L.
Exercise 1.3.1. Determine which links in Table 1.1 are alternating links. Record your findings. (Note that there are many nonalternating diagrams that can be created of a given alternating link, so just because a particular diagram is not alternating does not imply that no diagram of that link is alternating!)
Exercise 1.3.2. Choose one of the alternating links you found in Exercise 1.3.1 that has an alternating diagram in Table 1.1. Produce an equivalent, nonalternating diagram of this link.
Exercise 1.3.3. Consider the knot projection in Figure 1.2.2. Show that the precrossings can be resolved in such a way that the diagram becomes an alternating diagram. Investigate whether this can always be done for any knot projection. Can you find a knot projection for which there is no possible selection of crossings that will result in an alternating diagram? Record your findings, arguments, and conjectures.
The following three exercises relate to finding the crossing number of an alternating diagram.
Exercise 1.3.4. Consider the alternating diagrams in Table 1.1. For each alternating diagram, can you produce an equivalent diagram for this link that has fewer crossings? If so, is the resulting diagram alternating? Record your findings.
Exercise 1.3.5. Consider the alternating knots in Figure 1.3.1. For each alternating knot, can you produce another diagram for this knot that has fewer crossings? If so, is the resulting diagram alternating? Record your findings.
Exercise 1.3.6. Suppose K is an alternating knot. Use the exercise above to state a conjecture regarding any diagram of K that has a minimum number of crossings. Are there required properties for the diagram of K? Will a diagram of K with minimum crossings always be alternating? Create your own new examples to give further evidence in support of your conjecture.
We will revisit your conjecture and provide a proof determining the crossing number for alternating links in Chapter 6.
1.4 Games with Knots
Given a projection of a knot, we can play a game called the Knotting-Unknotting game. Suppose two players, Kenya and Ulysses, take the knot projection in Figure 1.4.1 as their starting "game board."
The players take turns resolving precrossings until all crossing information has been determined. Once a crossing has been resolved, it cannot be changed. Kenya's goal is to create a nontrivial knot, while Ulysses' goal is to make the unknot.
Exercise 1.4.1. Find a friend and decide who will play the part of Kenya and who will play the part of Ulysses. Next, decide who will play first and who will play second. Play the game on the projection in Figure 1.4.1. Who won? Did the winner seem to have an unfair advantage?
Exercise 1.4.2. Play the same game as in the previous exercise, but switch who goes first. Who won? Did the winner seem to have an unfair advantage?
Exercise 1.4.3. Now draw your own projection and play another round of the Knotting-Unknotting game. Describe any strategies you uncover for how each player should play the game.
Exercise 1.4.4. Is there a knot projection on which Ulysses (the unknotter) can always win, regardless of whether he plays first or second?
Exercise 1.4.5. Is there a knot projection on which Kenya (the knotter) can always win, regardless of whether she plays first or second?
Exercise 1.4.6. Invent and explore your own game that can be played on a knot diagram or a projection of a knot diagram.
1.5 Mirrors, Orientation & Inverses
The mirror image of a link diagram D, denoted [Dm, is the link obtained by changing all of the crossings of D. In other words, each crossing over-strand becomes the under-strand of that crossing and vice versa.
Exercise 1.5.1. Let D be a link diagram and D* be obtained by reflecting D through a mirror. Explain why D* and Dm result in equivalent links. (Use the description of equivalence of knots and links that was given at the beginning of this chapter.)
A link is called chiral if it is not equivalent to its mirror image. If it is equivalent to its mirror image, it is called amphichiral or achiral.
Exercise 1.5.2. Investigate whether the knots (c), (d), (n), and (o) in Table 1.1 are chiral or achiral. Record your findings and conjectures.
A link can be given an orientation by simply assigning a direction of travel around each loop. Orientation is typically indicated by drawing one or more small arrowheads as shown in Figure 1.5.2.
For an oriented knot diagram D, the same diagram with the opposite orientation is called the reverse of D, denoted . For some links, the choice of orientation does not matter. That is, the oriented link is equivalent to its reverse. A link with this property is called invertible.
Exercise 1.5.3. (1) Determine whether or not the links in Table 1.1 are invertible. (2) Create your own knot with seven or fewer crossings and determine whether or not your knot is invertible.
1.6 Knot Composition & Prime Knots
Given two link diagrams, we can create a new link by removing a small arc from each diagram and then connecting the four endpoints by two new arcs, as in Figure 1.6.1. For two links J and K, the new link is called the composition of J and K and is denoted J#K.
A knot is called a composite knot if it can be expressed as the composition of two nontrivial knots. The knots that make up the composite knot are called the factor knots.
Exercise 1.6.1. Determine the result of composing the unknot with a link L. Record your findings.
If a knot is not the composition of any two nontrivial knots, then it is called a prime knot.
Exercise 1.6.2. The knots in Figure 1.6.2 are composite. Make each knot out of string. Play with your physical knots to identify their prime factor knots.
(Continues...)
Excerpted from An Interactive Introduction to Knot Theory by Inga Johnson, Allison Henrich. Copyright © 2017 Inga Johnson and Allison Henrich. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Notes 6
1 Playing & Building Intuition 7
1.1 Projections, Diagrams & Equivalence 11
1.2 Crossing and Unknotting Numbers 12
1.3 Alternating Knots 14
1.4 Games with Knots 16
1.5 Mirrors, Orientation & Inverses 17
1.6 Knot Composition Prime Knots 18
1.7 Knot Notation 20
1.8 Questions in Knot Theory 22
2 Knot Definition & Equivalence 25
2.1 Polygonal Curves & Δ-Equivalence 26
2.2 Diagram Equivalence via R-Moves 29
2.3 The Equivalence of Δ- and R-Equivalence 39
2.4 Nonequivalence and Invariants 42
3 Families of Links and Braids 45
3.1 Twist Knots 46
3.2 Pretzel Links 47
3.3 Torus Links 48
3.4 Closed Braids 50
4 Knot Notation 63
4.1 DT Notation 64
4.2 Gauss Codes & Gauss Diagrams 67
4.3 Rational Knots & Conway Notation 72
5 Combinatorial Knot Invariants 95
5.1 The Writhe of a Diagram 96
5.2 The Linking Number 98
5.3 Tricolorability 101
5.4 A Generalization of Tricolorability 103
5.5 Matrices, Colorings & Determinants 107
6 Knot Polynomials 121
6.1 The Alexander Polynomial 122
6.2 The Kauffman Bracket & Jones Polynomial 129
6.3 Tail's Conjecture 134
7 Unknotting Operations & Invariants 143
7.1 Unknotting Operations 144
7.2 The Unknotting Number 158
7.3 The Region Unknotting Number 161
8 Virtual Knots 165
8.1 What is a Virtual Knot? 166
8.2 Virtual Knot Invariants 170
8.3 Virtual Unknotting 177
Acknowledgments 182
Index 183
Bibliography 187