Knot Theory and Its Applications

349 Pages·1996·17.507 MB·English

by  Kunio Murasugi (auth.)

Modern Blrkhauser Classics
Many ofthe original research and survey monographs in pure and
applied mathematics published by Blrkhauser in recent decades
have been groundbreaking and havecome to be regarded as foun-
dationalto the subject.Through the MBCSeries,aselect numberof
thesemodernclassics,entirelyuncorrected,arebeingre-releasedin
paperback(andaseBooks)to ensurethatthesetreasuresremainac-
cessible to newgenerations ofstudents, scholars, and researchers.
Knot Theory
&
Its Applications
Kunio Murasugi
Reprint ofthe 1996 Edition
Birkhauser
Boston • Basel • Berlin
KunioMurasugi
DepartmentofMathematics
UniversityofToronto
Toronto.OntarioM5S2E4
Canada
Originallypublishedasamonograph.
CoverdesignbyAlexGerasev.
MathematicsSubjectClassification(2000):57M25.57-01
LibraryofCongressControlNumber:2007933901
ISBN-13:978-0-8176-47I8-6 e-ISBN-13:978-0-8176-4719-3
Printedonacid-freepaper.
$ ~
Birkhiiuser
<92008BirkhauserBoston
Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit-
tenpermissionofthepublisher(BirkhauserBoston.cloSpringerScience-j-BusinessMediaLLC.233
SpringStreet.NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor
scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval.electronic
adaptation,computersoftware.orbysimilarordissimilarmethodologynowknownorhereafterde-
velopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms.evenifthey
arenotidentifiedasSUCh.isnottobetakenasanexpressionofopinionastowhetherornottheyare
subjecttoproprietaryrights.
9 8 765 4 3 2 I
www.birkhauser.com (lBT)
Kunia Murasugi
Translated by Bohdan Kurpita
KNOT THEORY
and
ITS APPLICATIONS
Birkhauser
Boston • Basel • Berlin
KunioMurasugi Bohdan Kurpita
DepartmentofMathematics The Daiwa Anglo-Japanese
University ofToronto Foundation and Waseda University
Toronto, Ontario M5S 1A1 Shinjuku-ku,Tokyo
Canada Japan
LibraryofCongressCataloging-in-PublicationData
Murasugi,Kunio,1929-
[Musubimerirontosonooyo. English]
Knottheoryanditsapplications I KunioMurasugi ; translatedby
BohdanKurpita.
p. em.
Includesbibliographicalreferences (p. -)andindex.
ISBN0-8176-3817-2 (alk.paper). -- ISBN3-7643-3817-2 (alk.
paper)
1. Knottheory. I. Title.
QA612.2.M8613 1996 96-16329
514'.224--dc20 CIP
Printedonacid-freepaper i5
Publishedoriginallyin1993inJapanese Birkhduser
©1996BirkhauserBoston
CopyrightisnotclaimedforworksofU.S.Governmentemployees.
Allrightsreserved. Nopartofthispublicationmaybereproduced,storedinaretrieval
system,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopy-
ing,recording,orotherwise.withoutpriorpermissionofthecopyrightowner.
Permissiontophotocopyforinternalorpersonaluseofspecificclientsisgrantedby
BirkhauserBostonforlibrariesandotherusersregistered withtheCopyrightClearance
Center(CCC),providedthatthebasefeeof$6.00percopy,plus$0.20perpageispaid
directlytoCCC,222RosewoodDrive.Danvers, MA01923,U.S.A. Specialrequests
shouldbeaddresseddirectlytoBirkhauserBoston.675MassachusettsAvenue,Cam-
bridge,MA02139,U.S.A.
ISBN0-8176-3817-2
ISBN3-7643-3817-2
Typesetin TEXbyBohdanKurpita
PrintedandboundbyMaple-Vail.York,PA
PrintedintheU.S.A.
9 8 7 6 5 4 3 2
c,,,,,,,,.
Introduction 1
FundamentalConceptsofKnotTheory 5
1 Theelementaryknotmoves 6
2 Theequivalenceofknots (I) 7
3 Theequivalenceofknots (II) 9
4 Links 14
5 Knotdecompositionandthesemi-groupofaknot 17
6 Thecobordismgroupof knots 23
KnotTables 25
1 Regulardiagramsandalternatingknots 26
2 Knot tables 30
3 Knotgraphs 34
FundamentalProblemsofKnotTheory 40
1 Globalproblems 41
2 Localproblems 43
ClassicalKnot Invariants 47
1 TheReidemeistermoves 48
2 The minimumnumberofcrossingpoints 56
3 The bridgenumber 58
4 The unknottingnumber 61
5 Thelinkingnumber 64
6 Thecolouringnumberofaknot 69
SeifertMatrices 75
1 The Seifertsurface 76
2 Thegenusofaknot 80
3 The Seifert matrix 83
4 S-equivalenceofSeifertmatrices 89
InvariantsfromtheSeifertmatrix 104
1 TheAlexanderpolynomial 105
2 TheAlexander-Conwaypolynomial 108
3 BasicpropertiesoftheAlexanderpolynomial 116
4 Thesignatureofaknot 122
TorusKnots 132
1 Torus knots 133
2 Theclassificationoftorusknots (I) 137
3 The Seifertmatrixofatorusknot 141
4 Theclassificationoftorusknots (II) 143
5 Invariantsoftorusknots 148
CreatingManifoldsfrom Knots 152
1 Dehn surgery 154
2 Coveringspaces 159
3 Thecycliccoveringspaceofaknot 163
4 AtheoremofAlexander 166
Tanglesand2.BridgeKnots 171
1 Tangles 172
2 Trivial tangles(rational tangles) 176
3 2-bridgeknots (rational knots) 182
4 Oriented2-bridgeknots 194
TheTheoryofBraids 197
1 Braids 198
2 The braidgroup 201
3 Knotsandbraids 209
4 Thebraidindex 214
TheJonesRevolution 217
1 TheJones polynomial 219
2 The basiccharacteristicsoftheJones polynomial 222
3 Theskeininvariants 231
4 TheK1.luffman polynomial 232
5 Theskeinpolynomialsandclassicalknotinvariants.
(AlternatingknotsandtheTait conjectures) 241
Knots via StatisticalMechanics 248
1 The6-vertexmodel 249
2 Thepartitionfunctionforbraids 255
3 Aninvariantofknots 260
Knot Theoryin MolecularBiology 267
1 DNAandknots 268
2 Site-specificrecombination 271
3 A modelforsite-specificrecombination 273
4 RecombinationduetotherecombinaseTn3Resolvase 276
GraphTheoryAppliedto Chemistry 284
1 Aninvariantofgraphs:thechromaticpolynomial 286
2 Bing'sconjectureandspatialgraphs 289
3 Thechiralityofspatialgraphs 296
VassilievInvariants 300
1 Singularknots 301
2 Vassiliev invariants 304
3 SomeexamplesofVassiliev invariants 308
4 Chorddiagrams 313
5 Final remarks 321
Appendix 325
Appendix(I):atableofknots 326
Appendix(II):AlexanderandJones polynomials 327
Notes 329
Bibliography 333
Index 337
Knots and braids have been extremely beneficial through the ages
to our actual existence and progress. For example, in the primordial
ages of our existence, in order to construct an axe a piece of stone
was bound/knotted to a sturdy piece of wood. To make a net, vines
or creepers, animal hair, et cetera were bound/braided together. Also
it is known that the ancient Inca civilization developed a system of
characters that were formed from knotted pieces ofstring. .
Although people have been making use of knots since the dawn of
ourexistence, theactual mathematicalstudyofknots isrelativelyyoung,
closer to 100 years than 1000 years. In contrast, Euclidean geometry
and number theory,which havebeen studiedovera considerable number
ofyears, germinated because of the cultural "pull" and thestrong effect
that calculations and computations generated. It is still quite cornmon
to see buildings with ornate knot or braid lattice-work. However, as
a starting point for a study of the mathematics of a knot, we need to
excoriate this aesthetic layer and concentrate on the shape of the knot.
Knot theory, in essence, is the study of the geometrical apects of these
shapes. Not only has knot theory developed and grown over the years
in its own right, but also the actual mathematics of knot theory has
been shown to have applications in various branchesofthesciences, for
example, physics, molecular biology,chemistry, et cetera.
In this book, we aim to guide the reader over the multifarious as-
pects that make up this theory ofknots. Weshall, in a straightforward
manner, explain the various concepts that form this theory of knots.
Throughout this book, we shall concentrate on lucid exposition, and
the exercises that can be found liberally sprinkled within act as a con-
duit between the theory and the understanding of this theory by the
reader. Therefore, this book is not just another book for those who
work or intend to work within the confines of knot theory, but is also
for those engaged in other areas in which knot theory may be applied
even if they do not have a considerable background in mathematics.
The general reader is also welcome, hopefully adding to the diversity
of knot theory. We shall cover what exactly knot theory is; what are
its motivations; its known results and applications; and what has been
discovered but is not yet completely understood.
Knot theory is a branch of the geometry of 3 dimensions. Since
three dimensions is the limit ofwhat is usually perceived intuitively, we
can call on this to help us explain concepts. To this end, in this book
we make extensive use of the numerous diagrams. Moreover, often the
intuitive approach is carried through into the actual text. However, the
proofs are still proven to the usual standard ofmathematical rigour. In
certaincases, fortheconvenienceofthereader, wehave appendedat the
end of this book several short, more detailed notes and commentaries.
Since we have tried as much as possible to avoid formal terminol-
ogy, i.e., we do not use concepts that are common in topology, such
as knot group and homology group, it has been necessary to leave out
several theorems and proofs. For the reader who is interested in a more
formal approach, a good guide is the book by Crowell and Fox [CF*].
For those interested in obtaining an even deeper understanding of knot
theory than that which may be garnered by reading this book only, we
recommend theresearch-level book edited by Kawauchi [K*]. Since the
purpose of the bibliography at the end of this book is to cite the theo-
rems that appear in the text, as a general bibliography for knot theory
it is inadequate. However, in Kawauchi [K*] and Burde and Zieschang
[BZ*] there can be found exhaustive bibliographies, so the inquisitive
reader who requires further references should consider consulting these
two bibliographies.
As asupplement,weinclude the knot diagrams ofprimeknots with
up to eight crossing points (35 in total), and a second table lists their
Alexander and Jones polynomials. Hopefully, this willprove ofpractical
use to the reader.
Finally, during the gestation period of this book I received the
valuable opinions of M. Sakuma, M. Saito and S. Yamada. Also, sev-
eral people kindly explained ideas to me that are outside my field of
speciality. The students of M. Sakuma and S. Suzuki provided many
additional, helpful comments about the original Japanese edition. To
all these individuals I express my deep gratitude. Furthermore, for the
Japanese language edition of this book I received much help from the
editorial staff of the publishers, especially from T. Kamei. For the En-
glish edition, the staff at Birkhauser in Boston, especially E. Beschler,
have been extremely helpful. To all these people I express my warmest
thanks.
Postscriptum, even in the fewyears since the Japanese version was pub-
lishedin1993therehavebeen interestingdevelopmentsinknottheory. In this
English translation,wehaveincorporatedsome ofthese recentdevelopments.