Other Title s i n Thi s Serie s

6 Davi d W . Farme r an d Theodor e B . Stanford , Knot s and surfaces : A guide t o discovering mathematics , 1996

5 Davi d W . Farmer , Group s and symmetry: A guide to discovering mathematics, 1996 4 V . V . Prasolov , Intuitiv e topology, 1995 3 L . E . Sadovski l an d A . L . SadovskiT , Mathematic s and sports , 1993 2 Yu . A . Shashkin , Fixe d points, 1991 1 V . M . Tikhomirov , Storie s about maxim a and minima , 1990

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Mathematical World • Volum e 6

Knots an d Surfaces

A Guide to Discoverin g Mathematics

David W. Farmer and

Theodore B . Stanfor d

American Mathematical Societ y

http://dx.doi.org/10.1090/mawrld/006

2000 Mathematics Subject Classification. Primar y 57-01 ; Secondar y 05-01 .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Farmer, Davi d W. , 1963 -Knots an d surface s : a guid e t o discoverin g mathematic s / Davi d W . Farmer , Theodor e B .

Stanford. p. cm . — (Mathematica l world , ISS N 1055-9426 ; v . 6 )

Includes bibliographica l references . ISBN 0-8218-0451- 0 (alk . paper ) 1. Grap h theory . 2 . Kno t theory . 3 . Surfaces . I . Stanford , Theodor e B. , 1964- . II . Title .

III. Series .

QA166.F37 199 5 511'.5—dc20 95-3089 7

CIP

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Table of Content s

Chapter 1 . Network s 1 1.1 Countrie s o f the insect worl d 1.2 Notation , an d a catalog 1.3 Tree s 1.4 Tree s in graphs 1.5 Euler' s formul a 1.6 Plana r graph s 1.7 Path s in graphs 1.8 Dua l graph s 1.9 A ma p of the United State s

1.10 Colorin g graphs 1.11 Th e six-color theore m 1.12 Note s

Chapter 2 . Surface s 31 2.1 Th e shape o f the world 2.2 Th e fla t toru s 2.3 Graph s on the toru s 2.4 Euler' s formula, agai n 2.5 Regula r graph s 2.6 Mor e surfaces: hole s 2.7 Mor e surfaces: connecte d sum s 2.8 One-side d surface s 2.9 Identifyin g two-side d surface s

2.10 Cel l complexe s 2.11 Note s

Chapter 3 . Knot s 5 9 3.1 Th e vie w from outsid e 3.2 Manipulatin g knot s 3.3 Lot s of knot s 3.4 Alternatin g knot s 3.5 Unknottin g numbe r 3.6 Link s 3.7 Linkin g numbe r

v

VI TABLE OF CONTENT S

3.8 Colorin g knot s an d link s 3.9 Note s

3.10 Catalo g o f knot s

Chapter 4 . Project s 8 7 4.1 Synthesi s Project s

Surfaces a s sculptur e Practical knot s Celtic knotwor k The famil y tre e of knot s Graphs, knots , and surface s i n ar t Aesthetics o f graphs, knots , and surface s Illustrate a mathematica l topi c Graphs, knots , and chemistr y

4.2 Mathematica l Project s Games on surface s The Doubl e toru s One-sided surface s Space Tyran t maze s Covering graph s Knots an d graph s Knot polynomial s

Bibliography 9 7

Index 99

Preface

This book is a guide to discovering mathematics. Every mathematic s textboo k i s filled with result s an d technique s whic h

once were unknown. Th e results were discovered by mathematicians who exper-imented, conjectured, discussed their work with others, and then experimented some more. Man y promising ideas turned out to be dead-ends, and lots of hard work resulted in little output. Ofte n th e first progress was the understanding of some special cases. Continue d work led to greater understanding, and some-times a complex picture began to be seen as simple and familiar. B y the time the work reaches a textbook, it bears no resemblance to its early form, and the details of its birth and adolescence have been lost. Th e precise and methodical exposition of a typical textbook is often the first contact one has with the topic, and this leads many people to mistakenly think that mathematics is a dry, rigid, and unchanging subject.

We believe tha t th e mos t excitin g par t o f mathematics i s the proces s of invention and discovery. Th e aim of this book is to introduce that proces s to you, th e reader. B y mean s of a wide variety of tasks, thi s book will lead you to discover some real mathematics. Ther e are no formulas to memorize. Ther e are no procedures to follow. B y looking at examples, searching for patterns in those examples, and then searching for the reasons behind those patterns, you will develop your own mathematical ideas . Th e book is only a guide; its job is to start you in the right direction , and to bring you back if you stray too far. The discovery is left to you.

This book is suitable for a one semester course at the beginning undergrad-uate level. Ther e are no prerequisites. An y college student interested in discov-ering the beauty of mathematics can enjoy a course taught from this book. An interested high school student will find this book to be a pleasant introduction to some modern areas of mathematics.

While preparin g thi s boo k w e were fortunate t o hav e access to excellen t notes taken by Hui-Chun Lee. We thank Klaus Peters and Gretchen Wright for helpful comments on an early version of this book.

David W. Farmer Theodore B. Stanford

September, 1995

VII

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Bibliography

[AC] Applied Combinatorics, b y Alan Tucker , Joh n Wile y & Sons, 1995 . A goo d introductio n t o combinatorics . Ha s a reasonabl e expositio n o f the Poly a enumeratio n formula . A variatio n o f tha t formul a ca n b e used t o coun t th e graphs wit h a given numbe r o f vertices .

[E] Ethnomathematics: A Multicultural View oi Mathematical Ideas, b y Mar -cia Ascher , Brooks/Cole , 1991.

Presents a n interestin g accoun t o f th e mathematica l sophisticatio n o f 'primitive' people .

[F] Flatland: a romance in many dimensions, b y Edwi n Abbott , Dove r Books . An amusin g stor y o f a 3-dimensiona l bein g wh o visit s a 2-dimensiona l world. Mor e socia l commentar y tha n mathematics . A n excellen t up -date, withou t th e socia l commentary , appear s i n The Shape of Space.

[GS] Groups and Symmetry: a guide to discovering mathematics, b y Davi d W. Farmer , America n Mathematica l Society , 1996 .

A boo k i n th e sam e styl e a s Knots and Surfaces.

[GT] Graph Theory, b y Fran k Harary , Addison-Wesley , 1969 . A classi c introductor y boo k o n grap h theory . I t begin s wit h lot s o f definitions, s o don' t hesitat e t o ski p t o th e middl e an d the n loo k bac k for th e term s yo u need .

[KB] The Knot Book, b y Coli n Adams , W.H . Freeman , 1994 . An introductio n t o knots . Th e boo k elaborate s o n man y o f the topic s briefly mentione d i n Knots and Surfaces. A goo d sourc e o f idea s fo r further study . Mention s man y unsolve d problems .

[MS] Mathematical Snapshots, b y Hug o Steinhaus , variou s publishers . Short chapter s o n a variet y o f mathematics , writte n fo r a genera l au -dience. Al l o f i t i s interesting , an d tw o o r thre e o f th e chapter s ar e relevant t o thi s book .

[SoS] The Shape of Space, b y Jeffrey Weeks , Marce l Dekker , 1985 . An excellen t book . I f yo u like d Knots and Surfaces, the n yo u shoul d be abl e t o understan d an d appreciat e mos t o f The Shape of Space.

97

98 BIBLIOGRAPHY

[SSS] Shapes , Space, and Symmetry, b y Alan Holden, Dover Books, 1991. Pictures of hundreds of 3-dimensional symmetric shapes. Lot s of regu-lar and semi-regular solids.

[VM] The Visual Mind, Michele Emmer, Ed., MIT Press, 1993. A collection of 36 papers dealing with mathematical aspects of art. A few are relevant to this book. Severa l pictures of interesting sculpture.

Index

alternating diagram, 6 9 alternating knot , 6 9 annulus, 43

Borromean rings , 7 3 boundary, 4 6

cell, 38 cell complex , 5 3 X, chi, 5 6 circuit, 7 closed path, 7 closed surface , 9 0 coloring

knot, 8 0 link, 82

component, 5 , 7 3 composite knot , 6 8 connected graph , 4 , 7 connected sum , 44, 6 8

knots, 6 8 surfaces, 4 4

Connors, Jimmy , 5 contractible loop , 3 2 covering graph , 9 3 crosscap, 9 1 crossing, 6 0 crossing elimination , 7 4 crossing number , 6 4 cube, 42 cyclic graph, 4 0 cylinder, 4 3

decagon, 5 4 digraph, 2 8 directed graph, see digraph disconnected graph , 4

disk, 4 3 dodecagon, 5 4 dodecahedron, 4 2 double o f a link, 74 double torus , 45 dual graph , 1 6

edge, 4 equivalent, 4 4

knot diagram , 60 surface, 5 0

Euler characteristic , 5 6 Euler circuit , 1 4 Euler path, 1 4 Euler's formul a

plane, 1 0 projective plane, 9 1 sphere, 3 7 torus, 38

Euler, Leonhard , 10 , 27 pronunciation, 1 0

evil galacti c tyrant , 9 2

faces o f a graph, 1 1 family o f knots, 7 2 figure-8 knot , 65 flat torus , 35 four-color theorem , 2 0

galactic tyrant , 92 genus, 5 6 graph, 4, 5

coloring, 21 component, 5 connected, 4 , 7 covering, 9 3 cyclic, 40

99

100 INDEX

diagram, 4 disconnected, 4 dual, 1 6 faces, 1 1 path in , 6 Petersen, 2 9 planar, 1 0 regular, 3 9 signed, 9 4

Hamilton, Si r William Rowan , 1 6 Hamiltonian circuit , 1 6 Hamiltonian path , 1 6 handshake principle , 1 6 Hopf link , 7 3

icosahedron, 4 2

King Solomon' s knot , 7 3 Klein bottle , 48 , 91 Klein, Felix , 4 8 Kn, 1 0

K7 o n Klei n bottle , 9 2 K5

not planar , 1 2 on Klei n bottle , 4 9 on projectiv e plane , 4 9 on torus , 3 6

K^3 no t planar , 1 3 knot, 5 9

alternating, 6 9 coloring, 8 0 composite, 6 8 crossing number , 6 4 diagram, 5 9 figure-8, 6 5 mirror image , 6 6 oriented, 6 5 prime, 6 8 projection, 6 4 strand, 6 0 trefoil, 6 5 twist, 7 1 unknot, 6 2 unknotting number , 7 1

knot diagram , 5 9 Kuratowski's theorem , 2 9

left trefoil , 6 5 line, 4 link, 7 3

Borromean rings , 7 3 coloring, 8 2 component, 7 3 double of , 7 4 Hopf, 7 3 King Solomon' s knot , 7 3 splittable, 7 4 Whitehead, 7 3

linking number , 7 7

map coloring , 2 0 maze

space tyrant , 9 2 mirror imag e o f a knot , 6 6 Mobius strip , 4 7 Mobius , August, 4 7 multiple edges , 5

nonorientable surface , 5 7

octagon, 5 4 octahedron, 4 2 one-sided surface , 47 , 90 order, 11 , 15

face, 1 1 vertex, 1 5

orientable surface , 5 7 oriented knot , 6 5

pair o f pants , 4 3 path, 6

closed, 7 simple, 7

Petersen graph , 2 9 planar diagram , 1 0 planar graph , 1 0 point, 4 prime knot , 6 8 projection, 6 4 projective plane , 4 8

regular graph , 3 9 regular solid , 4 2 Reidemeister moves , 7 7 Reidemeister, Kurt , 7 7 right trefoil , 6 5

sculpture, 8 7 seven bridge s o f Konigsber g , 27 signed graph , 9 4 simple path , 7 Six Degrees o f Separation , 5 space tyrant , 9 2 spanning tree , 8 sphere, 3 1 splittable link , 7 4 sprouts, 8 9 strand, 6 0 subgraph, 2 8 surface

closed, 9 0 crosscap, 9 1 equivalent, 5 0 nonorientable, 5 7 one-sided, 47 , 90 orient able, 5 7 standard form , 5 8

tennis, 5

INDEX 10 1

tetrahedron, 4 2 topology, 5 1 tori, 5 6 torus, 3 1

flat, 3 5 hexagon, 3 7

tree, 7 trefoil, 6 5 triangulation, 5 5 triple torus , 4 5 twist knot , 7 1 tyrant

evil galactic , 9 2

unknot, 6 2 unknotting number , 7 1

valence, 2 8 vertex, 4

order, 1 5 vertices, see vertex

Whitehead link , 7 3