Table Of ContentColin C. Adams
II
W. H. Freeman and Company
New York
Contents
Preface xi
Chapter 1
Introduction 1
1.1 Introduction 1
1.2 Composition of Knots 7
1.3 Reidemeister Moves 12
1.4 Links 16
1.5 Tricolorability 22
1.6 Knots and Sticks 27
Chapter 2
Tabulating Knots 31
2.1 History of Knot Tabulation 31
2.2 The Dowker Notation for Knots 3!:
2.3 Conway's Notation 41
2.4 Knots and Planar Graphs 51
viii Contents
Chapter3
Invariants of Knots 57
3.1 Unknotting Number 57
3.2 Bridge Number 64
3.3 Crossing Number 67
Chapter4
Surfaces and Knots 71
4.1 Surfaces without Boundary 71
4.2 Surfaces with Boundary 87
4.3 Genus and Seifert Surfaces 95
Chapter5
Types of Knots 107
5.1 Torus Knots 107
5.2 Satellite Knots 115
5.3 Hyperbolic Knots 119
5.4 Braids 127
5.5 Almost Alternating Knots 139
Chapter6
Polynomials 147
6.1 The Bracket Polynomial and the Jones Polynomial 147
6.2 Polynomials of Alternating Knots 156
6.3 The Alexander and HOMFLY Polynomials 165
6.4 Amphicheirality 176
Chapter?
Biology, Chemistry, and Physics 181
7.1 DNA 181
7.2 Synthesis of Knotted Molecules 195
7.3 Chirality of Molecules 201
7.4 Statistical Mechanics and Knots 205
Contents
Chapter8
Knots, Links, and Graphs 215
8.1 Links in Graphs 215
8.2 Knots in Graphs 222
8.3 Polynomials of Graphs 231
Chapter 9
Topology 243
9.1 Knot Complements and Three-Manifolds 243
9.2 The Three-Sphere and Lens Spaces 246
9.3 The Poincare Conjecture, Dehn Surgery, and the
Gordon-Luecke Theorem 257
Chapter 10
Higher Dimensional Knotting 265
10.1 Picturing Four Dimensions 265
10.2 Knotted Spheres in Four Dimensions 272
10.3 Knotted Three-Spheres in Five-Space 273
Knot Jokes and Pastimes 216
Appendix: 219
Table of Knots, Links, and Knot and Link Invariants
Suggested Readings and References 291
Index 303
Preface
Mathematics is an incredibly exciting and creative field of endeavor. Yet
most people never see it that way. Nonmathematicians too often assume
that we mathematicians sit around talking about what Newton did three
hundred years ago or calculating a couple of extra million digits of
'TT.
They do not realize that more new mathematics is being created now than
at any other time in the history of humankind.
Explaining the field of knot theory is a particularly effective way to
dispel this misconception. Here is a field that is over one hundred years
old, and yet some of the most exciting results have occurred in the last fif
teen years. Easily stated open questions still abound, and one can get a
taste for what it is like to do research very quickly. The other tremendous
advantage that knot theory has over many other fields of mathematics is
that much of the theory can be explained at an elementary level. One does
not need to understand the complicated machinery of adva:i;tced areas of
mathematics to prove interesting results.
My hope is that_ this book will excite people about mathematics-that
it will motivate them to continue to explore other related areas of mathe
xi
xii Preface
matics and to proceed to such topics as topology, algebra, differential ge
ometry, and algebraic topology.
Unfortunately, mathematics is often taught as if the only goal were to
pass a body of information from one person to the next. Although this is
certainly an important goal, it is essential to teach an appreciation for the
beauty of mathematics and a sense of the excitement of doing mathemat
ics. Once readers are hooked, they will fill in the details themselves, arid
they will go a lot farther and learn a lot more.
Who, then, is this book for? It is aimed at anyone with a curiosity
about mathematics. I hope people will pick up this book and start reading
it on their own. I also hope that they will do the exercises: the only way to
learn mathematics is to do it. Some of the exercises are straightforward;
others take some thought. The very hardest are starred and can be a bit
more challenging.
Scientists with primary interests in physics or biochemistry should
find the applications of knot theory to these fields particularly fascinating.
Although these applications have only been discovered recently, already
they have had a huge impact.
This book can be and has been used effectively as a textbook in
classes. With the exception of a few spots, the book assumes only a famil
iarity with high school algebra. I have also given talks on selected topics
from this book to high school students and teachers, college students, and
students as young as seventh graders.
The first six chapters of the book are designed to be read sequentially.
With the exception that Section 8.3 depends on Section 7.4, the remaining
four chapters are independent and can be read in any order. The topics
chosen for this book are not the standard topics that one would see in a
more advanced treatise on knot theory. Certainly the most glaring omis
sion is any discussion of the fundamental group. My desire to make this
book more interesting and accessible to an audience without advanced
background has p:J;"ecluded such topics.
The choice of topics has been made by looking for areas that are easy
to understand without much background, are exciting, and provide op
portunities for new research. Some of the topics such as almost alternating
knots are so new that little research has yet been done on them, leaving
numerous open questions.
Although I drew on many sources while writing this book, I relied
particularly heavily on the writings and approaches of Joan Birman, John
Conway, Cameron Gordon, Vaughan Jones, Louis Kauffman, Raymond
Lickorish, Ken Millett, Jozef Przytycki, Dale Rolfsen, Dewitt Sumners,
Morwen Thistlethwaite, and William Thurston.
I would like to thank all the following colleagues, who contributed in
numerable comments and suggestions during the writing of this book,
and corrected many of the mistakes therein: Daniel Allcock, Thomas Ban-
Preface xiii
choff, Timothy Bremer, Patrick Callahan, J. Scott Carter, Peter Cromwell,
Alan Durfee, Dennis Garity, Jay Goldman, Cameron Gordon, Joel Hass,
James Hoste, Vaughan Jones, Frank Morgan, Monica Nicolau, Jozef Przy
tycki, Joseph O'Rourke, Alan Reid, Yongwu Rong, Dewitt Sumners, Mor
wen Thistlethwaite, Abigail Thompson, and Jeffrey Weeks. The filmstrip
format employed in the figures in Chapter 10 originated with J. Scott
Carter.
I also would like to thank all the students who have contributed to
this book. I was originally motivated to write this book through my partic
ipation in the SMALL Undergraduate Research Project at Williams Col
lege. Each summer since 1988, between fifteen and twenty-five students
have come to Williams College to work on mathematical research with
five to eight faculty over a ten-week period, through funds provided by
the National Science Foundation, the New England Consortium for Un
dergraduate Science Education, Williams College, and other granting
agencies. My group of students has usually worked on knot theory. Every
summer, I would find myself teaching them the same material over again,
without a reference at the right level. It was such beautiful material that I
decided it would be worth writing a book. Although this list does not in
clude all the students who have contributed to the book, I would like to
thank the following: Aaron Abrams, Charene Arthur, Jeffrey Brock, Derek
Bruneau, John Bugbee, Elizabeth Camp, Mark Chrisman, Tim Comar, Tara
de Souza, Keith Faigin, Joseph Francis, Thomas Graber, Lisa Harrison,
Daniel Heath, Martin Hildebrand, Hugh Howards, Amy Huston, Anne
Joseph, Lisa Klein, Katherine Kollett, Joshua Kucera, John MacEachem,
Lothar Mans, John Mynttinen, David Pesikoff, Jessica Polito, Dan Robb,
William Sherman, John Terilla, Pinnarat Vongsinsirikul, and Edward
Welsh.
Additional help came from Jeremiah Lyons, Marissa Barschdorf, and
Pier Gustafson, Christine Hastings, and Mel Slugbate. Christine Heinitz
and Thomas Banchoff deserve special credit for their work on the illustra
tions.
Colin C. Adams
February 1994
Introduction
1 .1
Introduction
Take a piece of string. Tie a knot in it. Now glue the two ends of the string
together to form a knotted loop. The result is a string that has no loose
ends and that is truly knotted. Unless we use scissors, there is no way that
we can untangle this string. (See Figure 1.1.)
Figure 1.1 Forming a knot from a piece of string.
2 The Knot Book
A knot is just such a knotted loop of string, except that we think of the
string as having no thickness, its cross-section being a single point. The
knot is then a closed curve in space that does not intersect itself anywhere.
We will not distinguish between the original closed knotted curve and
the deformations of that curve through space that do not allow the curve
to pass through itself. All of these deformed curves will be considered to
be the same knot. We think of the knot as if it were made of easily de
formable rubber.
Figure 1.2 Deforming a knot doesn't change it.
In these pictures of knots (Figure 1.2) one section of the knot passes
under another section at each crossing. The simplest knot of all is just the
unknotted circle, which we call the unknot or the trivial knot. The next
simplest knot is called a trefoil knot. (See Figure 1.3.) But how do we
know these are actually different knots? How do we know that we
couldn't untangle the trefoil knot into the unknot without using scissors
and glue, if we played with it long enough?
0
a b
Figure 1.3 (a) The unknot. (b) A trefoil knot.
Certainly, if you make· a trefoil knot out of string and try untangling it
into the unknot, you will believe very quickly that it can't be done. But we
won't be able to prove it until we introduce tricoloration of knots in Sec
tion 1.5.
In the table at the back of the book, there are numerous pictures of
knots. All of these knots are known to be distinct. If we made any one of
them out of string, we would not be able to deform it to look like any
of the others. On the other hand, here is a picture (Figure 1.4) of a knot
Introduction 3
that is actually a trefoil knot, even though it looks completely different
from the previous picture of a trefoil.
Figure 1.4 A nonstandard picture of the trefoil knot.
exercise 1.1 Make this knot out of string and then rearrange it to show
that it is the trefoil knot. (Actually, an electrical extension cord works
better than string. You can tie a knot in it and then plug it into itself in
order to form a knot. A third option is to draw a sequence of pictures
that describe the deformation of the knot. This is particularly easy to
do on a blackboard, with chalk and eraser. As you deform the knot,
you can simply erase and redraw the appropriate sections of the pic
ture.)
There are many different pictures of the same knot. In Figure 1.5, we
see three different pictures of a new knot, called the figure-eight knot. We
call such a picture of a knot a projection of the knot.
Figure 1.5 Three projections of the figure-eight knot.
The places where the knot crosses itself in the picture are called the
crossings of the projection. We say that the figure-eight knot is a four
crossing knot because there is a projection of it with four crossings, and
there are no projections of it with fewer than four crossings.
If a knot is to be nontrivial, then it had better have more than one
crossing in a projection. For if it only has one crossing, then the four ends
of the single crossing must be hooked up in pairs in one of the four ways
