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Extensions of quandles and cocycle knot invariants

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Extensions of quandles and cocycle knot invariants
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Appiou Nikiforou, Marina
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knot theory
coloring
algebraic structures
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ABSTRACT: Knot theory has rapidly expanded in recent years. New representations of braid groups led to an extremely powerful polynomial invariant, the Jones polynomial. Combinatorics applied to knot and link diagrams led to generalizations. Knot theory also has connections with other fields such as statistical mechanics and quantum field theory, and has applications in determining how certain enzymes act on DNA molecules, for example. The principal objective of this dissertation is to study the relations between knots and algebraic structures called quandles. A quandle is a set with a binary operation satisfying some properties related to the three Reidemeister moves. The study of quandles in relation to knot theory was intitiated by Joyce and Matveev. Later, racks and their (co)homology theory were defined by Fenn and Rourke. The rack (co)homology was also studied by Grana from the viewpoint of Hopf algebras. Furthermore, a modified definition of homology theory for quandles was introduced by Carter, Jelsovsky, Kamada, Langford, and Saito to define state-sum invariants for knots and knotted surfaces, called quandle cocycle invariants. This dissertation studies the quandle cocycle invariants using extensions of quandles and knot colorings. We obtain a coloring of a knot by assigning elements of a quandle to the arcs of the knot diagram. Such colorings are used to define knot invariants by state-sum. For a given coloring, a 2-cocycle is assigned at each crossing as the Boltzmann weight. The product of the weights over all crossings is the contribution to the state-sum, which is the formal summation of the contributions over all possible colorings of the given knot diagram by a given quandle. Generalizing the cocycle invariant for knots to links, we define two kinds of invariants for links: a component-wise invariant, and an invariant defined as families of vectors. Abelian extensions of quandles are also defined and studied. We give a formula for creating infinite families of abelian extensions of Alexander quandles. These extensions give rise to explicit formulas for computing 2-cocycles. The theory of quandle extensions parallels that of groups. Moreover, we investigate the notion of extending colorings of knots using quandle extensions. In particular, we show how the obstruction to extending the coloring contributes to the non-trivial terms of the cocycle invariants for knots and links. Moreover, we demonstrate the relation between these new cocycle invariants and Alexander matrices.
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Thesis (Ph.D.)--University of South Florida, 2002.
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by Marina Appiou Nikiforou.
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ABSTRACT: Knot theory has rapidly expanded in recent years. New representations of braid groups led to an extremely powerful polynomial invariant, the Jones polynomial. Combinatorics applied to knot and link diagrams led to generalizations. Knot theory also has connections with other fields such as statistical mechanics and quantum field theory, and has applications in determining how certain enzymes act on DNA molecules, for example. The principal objective of this dissertation is to study the relations between knots and algebraic structures called quandles. A quandle is a set with a binary operation satisfying some properties related to the three Reidemeister moves. The study of quandles in relation to knot theory was intitiated by Joyce and Matveev. Later, racks and their (co)homology theory were defined by Fenn and Rourke. The rack (co)homology was also studied by Grana from the viewpoint of Hopf algebras. Furthermore, a modified definition of homology theory for quandles was introduced by Carter, Jelsovsky, Kamada, Langford, and Saito to define state-sum invariants for knots and knotted surfaces, called quandle cocycle invariants. This dissertation studies the quandle cocycle invariants using extensions of quandles and knot colorings. We obtain a coloring of a knot by assigning elements of a quandle to the arcs of the knot diagram. Such colorings are used to define knot invariants by state-sum. For a given coloring, a 2-cocycle is assigned at each crossing as the Boltzmann weight. The product of the weights over all crossings is the contribution to the state-sum, which is the formal summation of the contributions over all possible colorings of the given knot diagram by a given quandle. Generalizing the cocycle invariant for knots to links, we define two kinds of invariants for links: a component-wise invariant, and an invariant defined as families of vectors. Abelian extensions of quandles are also defined and studied. We give a formula for creating infinite families of abelian extensions of Alexander quandles. These extensions give rise to explicit formulas for computing 2-cocycles. The theory of quandle extensions parallels that of groups. Moreover, we investigate the notion of extending colorings of knots using quandle extensions. In particular, we show how the obstruction to extending the coloring contributes to the non-trivial terms of the cocycle invariants for knots and links. Moreover, we demonstrate the relation between these new cocycle invariants and Alexander matrices.
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OceofGraduateStudies UniversityofSouthFlorida Tampa,Florida CERTIFICATEOFAPPROVAL Thisistocertifythatthedissertationof MARINAAPPIOUNIKIFOROU inthegraduatedegreeprogramof Mathematics wasapprovedonDecember6,2002 fortheDoctorofPhilosophydegree ExaminingCommittee: MajorProfessor:MasahikoSaito,Ph.D. Member:W.EdwinClark,Ph.D. Member:Nata saJonoska,Ph.D. Member:DavidRabson,Ph.D. ProgramDirector CommitteeVerifcation:

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EXTENSIONSOFQUANDLESANDCOCYCLEKNOTINVARIANTS by MARINAAPPIOUNIKIFOROU Adissertationsubmittedinpartialful“llment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida DateofApproval: December6,2002 MajorProfessor:MasahikoSaito,Ph.D.

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c CopyrightbyMarinaAppiouNikiforou2002 Allrightsreserved

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DEDICATION Tomyparentsforhavingfaithinme

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ACKNOWLEDGMENTS IgratefullyacknowledgethehelpofMasahikoSaito,whohassupervisedmydissertationandguidedmethroughallstepsofresearchandwriting.Iamgratefulto EdwinClark,Nata saJonoska,andDavidRabsonfortheirvaluablequidanceand feedback.Ialsothankmyco-authorsJ.ScottCarter,MohamedElhamdadi,and AngelaHaris. Ihavereceivedinvaluablesupportfrommyhusband,SavvasNikiforou,whohas providedhelpandencouragementthroug hmygraduatestudies.Iamalsothankful toSavvasforhisfeedbackandhelpwithLAT E X.Finally,Iwouldliketothanktomy parents,AndreasAppiosandParaskeviA ndreou,fortheirsupportandunderstanding.

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TABLEOFCONTENTS LISTOFTABLESii LISTOFFIGURESiii ABSTRACTiv CHAPTER1INTRODUCTION1 1.1Historyandorganization1 1.2Knotsandlinks2 1.3Fox n -coloringofknotdiagrams7 CHAPTER2QUANDLES9 2.1De“nitions9 2.2Examplesofquandles12 2.3Classi“cationof4-elementquandles13 2.4Coloringsofknotdiagramsbyquandles20 CHAPTER3COHOMOLOGYTHEORYOFQUANDLESANDCOCYCLEKNOTINVARIANTS25 3.1Homologyandcohomologyofquandles25 3.2Cocycleknotinvariants27 3.3Variationsofcocycleknotinvariants29 CHAPTER4EXTENSIONSOFQUANDLESBY2-COCYCLES32 4.1Abelianextensions32 4.2Evaluationsofcocycleinvariantsusingextensioncocycles44 CHAPTER5EXTENDINGCOLORINGSOFKNOTS51 5.1Extensionsofcolorings51 5.2Cocycleknotinvariantsasobstructionstoextendingcolorings52 CHAPTER6RELATIONSTOALEXANDERMATRICES57 6.1CocycleinvariantsandAlexandermatrices57 6.2ArelationtoAlexander-Conwaypolynomial65 REFERENCES67 ABOUTTHEAUTHOREndPage i

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LISTOFTABLES Table1.Multiplicationtablesfor R4, P3 T1,and Y415 Table2.Comparisonof3-elementquandlesof Y4with P320 ii

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LISTOFFIGURES Figure1.Restrictionsoftheprojectiononlinesegments4 Figure2.Awellroundedtrefoil5 Figure3.Positiveandnegativecrossings6 Figure4.TheReidemeistermoves6 Figure5.The3-colorabilitycondition7 Figure6.Acoloredtrefoil8 Figure7.The n -colorabilitycondition8 Figure8.Reidemeistermovesandquandleconditions10 Figure9.Quandlerelationatacrossing21 Figure10.Atrefoilcoloredbythequandle R321 Figure11.Wirtingerrelationofthefundamentalquandle23 Figure12.Weightsforpositiveandnegativecrossings27 Figure13.TypeIIImoveandthequandleidentity28 Figure14.The(4,2)-toruslink29 Figure15.AcoloredWhiteheadlink44 Figure16.Whiteheadlink46 Figure17.Borromeanrings49 Figure18.Labelingacrossing58 iii

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EXTENSIONSOFQUANDLESANDCOCYCLEKNOTINVARIANTS by MARINAAPPIOUNIKIFOROU AnAbstract Ofadissertationsubmittedinpartialful“llment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida DateofApproval: December6,2002 MajorProfessor:MasahikoSaito,Ph.D. iv

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Knottheoryhasrapidlyexpandedinrecentyears.Newrepresentationsofbraid groupsledtoanextremelypowerfulpolynomialinvariant,theJonespolynomial. Combinatoricsappliedtoknotandlinkdiagr amsledtogeneralizations.Knottheory alsohasconnectionswithother“eldssu chasstatisticalmechanicsandquantum “eldtheory,andhasapplicationsindete rmininghowcertainenzymesactonDNA molecules,forexample. Theprincipalobjectiveofthisdissertationistostudytherelationsbetweenknots andalgebraicstructurescalledquandles .AquandleisasetwithabinaryoperationsatisfyingsomepropertiesrelatedtothethreeReidemeistermoves.Thestudy ofquandlesinrelationtoknottheorywasintitiatedbyJoyceandMatveev.Later, racksandtheir(co)homologytheorywerede“nedbyFennandRourke.Therack (co)homologywasalsostudiedbyGra nafromtheviewpointofHopfalgebras.Furthermore,amodi“edde“nitionofhomologytheoryforquandleswasintroducedby Carter,Jelsovsky,Kamada,Langford,andSaitotode“nestate-suminvariantsfor knotsandknottedsurfaces,calledquandlecocycleinvariants. Thisdissertationstudiesthequandlecocycleinvariantsusingextensionsofquandlesandknotcolorings.Weobtainacoloringofaknotbyassigningelementsof aquandletothearcsoftheknotdiagram.Suchcoloringsareusedtode“neknot invariantsbystate-sum.Foragivencoloring,a2-cocycleisassignedateachcrossingastheBoltzmannweight.Theproductoftheweightsoverallcrossingsisthe contributiontothestate-sum,whichistheformalsummationofthecontributions overallpossiblecoloringsofthegivenknotdiagrambyagivenquandle.Generalizing thecocycleinvariantforknotstolinks,wede“netwokindsofinvariantsforlinks:a component-wiseinvariant,andaninvariantde“nedasfamiliesofvectors. Abelianextensionsofquandlesarealsode“nedandstudied.Wegiveaformulafor creatingin“nitefamiliesofabelianexten sionsofAlexanderquandles.Theseextensionsgiverisetoexplicitformulasforcomputing2-cocycles.Thetheoryofquandle v

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extensionsparallelsthatofgroups.Moreo ver,weinvestigatethenotionofextending coloringsofknotsusingquandleextensi ons.Inparticular,weshowhowtheobstructiontoextendingthecoloringcontributestothenon-trivialtermsofthecocycle invariantsforknotsandlinks.Moreover,wedemonstratetherelationbetweenthese newcocycleinvariantsandAlexandermatrices. AbstractApproved: MajorProfessor:MasahikoSaito,Ph.D. AssociateProfessor,DepartmentofMathematics DateApproved: vi

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CHAPTER1 INTRODUCTION 1.1Historyandorganization Knottheoryisthemathematicalstudyofknots.Aknotisaclosed,non-intersecting curvein3-dimensionalEuclideanspace.Moreprecisely,aknotistheimageofan embeddingofaunitcircleintheEuclidean3-space.Muchofknottheoryisconcerned withtellingwhichknotsarethesameandwhicharedierent.Knottheoryhasbeen used,forexample,todeterminehowcertainenzymesactonDNAmolecules.Oneof theearlyandmainachievementswasthediscoveryin1923oftheAlexanderpolynomialofaknotorlink[1].Homologytheo ryappliedtoin“nitecycliccoversofthe complementofaknotledtotheAlexanderp olynomial.Conway[13]gavearecursive formulaofthispolynomial.Thestudyofknotinvariantschangeddramaticallyin 1984,whennewrepresentationsofbraidgroupsledtoanotherextremelypowerful polynomialinvariant,theJonespolynomi al[26].Sincethenmanygeneralizations werediscovered.Thesecausedinteractionsbetweenknottheoryandvariousother “eldssuchascombinatorics,statistic almechanics,andquantum“eldtheory. Thisdissertationconsistsoftwoparts.The“rstpart(Chapters1…3)isanoverview ofinvariantsofknotsandlinksde“nedbyusingquandles.Thesecondpart(Chapters 4…6),whichisthemaincontributionoftheauthor,dealswithextensionsofquandles andcolorings,aswellasrelationstoAlexandermatrices. InChapter1,wereviewbackgroundinformationneededtopresentthiswork. Constructionsofquandlesandsomeofthe irpropertiesarediscussedinChapter2. Inparticular,weconsidercolori ngsofknotdiagramsbyquandles. 1

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HomologyandcohomologytheoriesforquandlesareintroducedinChapter3. Generalizingthequandlecocycleinvarian tsforknots[8],wede“neacomponent-wise invariant,andaninvariantforlinksde“nedbyfamiliesofvectors. InChapter4,wediscussabelianextensionsofquandles.Formulasthatproducein“nitefamiliesofabelianextensionsofAlexanderquandlesaregiven,and thesefamiliesareshowntobenon-trivial asextensions.Forexample,weshowthat Zqm +1[ T,TŠ 1] / ( T Š 1+ q )isanabelianextensionofthequandle X = Zqm[ T,TŠ 1] / ( T Š 1+ q ),forsomecocycle Z2 Q( X ; Zq).Moreover,theseextensionsgiverisetoexplicit formulasforcomputingcocycles.InChap ter5,wedescribethenotionofextending coloringsofknotdiagramsusingtheexten siontheoryofquandles.Inaddition,we showhowthepreviouslyde“nedinvariant sdetermineforbothknotsandlinksthe numberofcoloringsbyaquandlethatcanbeextendedtocoloringsbyanextensionof thequandle.Finally,inChapter6werelatethenewcocycleinvariantstoAlexander matrices. 1.2Knotsandlinks Weoftendealwithknotsbydepictingtheminaplane;inotherwords,westudytheir diagrams.Moreover,wedescribetheequiva lenceofknotsbysomemovesamongtheir diagrams,calledtheReidemeistermoves.Tait[38]attemptedtoclassifyknottypes inthelate19thcentury. Oneofthemaintopicsofknottheoryisthestudyofknotinvariants.Aninvariant isatooltodistinguishknots.Itisawell-de“nedalgebraicobjectsuchasanumber, apolynomial,oragroup. Wedenotethe n -dimensionalEuclideanspaceby Rnandthe n -dimensionalsphere by Sn. 2

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Denition1.2.1 [30]A link L of m components isasubsetof S3,orof R3,that consistsof m disjoint,piecewiselinear,simpleclosedcurves.Alinkofonecomponent isa knot Denition1.2.2 [29]An orientation ofan n -simplexisanequivalenceclassoforderingsofthe n +1verticesmoduloevenpermutations.By[ v0,v1,...,vn],wedenote anorientedsimplexwithverticesorderedas v0,v1,...,vnandby Š [ v0,v1,...,vn],we denotethesimplexwithoppositeorientation.Bythe inducedorientation oftheface of[ v0,v1,...,vn]oppositeto vi,wemeantheorientationgivenby ( Š 1)i[ v0,v1,..., vi,...,vn] where vimeansthatthevertex i isommited.An orientation ofapiecewiselinear manifold M isanassignmentofanorientationforeach n -simplexof KM,atriangulationof M ,suchthattheorientationof A0inducedfromtheorientationof A1is oppositetothatof A0inducedfromtheorientationof A2forany n -simplices A1,A2in KM,where A0= A1 A2isan( n Š 1)-simplex.Accordingtowhetherornot suchanorientationexists,wesaythat M is orientable or non-orientable .When M isorientableandanorientationisspeci“ed, M issaidtobe oriented Since S1isa1-manifold,itisorientable,andsoareknotsandlinks. Denition1.2.3 [30]Links L1and L2in S3are equivalent ifthereisanorientationpreservingpiecewiselinearhomeomorphism h : S3 S3suchthat h ( L1)=( L2). AsimplewaytostudylinksistoworkwiththeirdiagramsŽ,whicharetwodimensionalrepresentationswith respecttothestandardprojection p : R3 R2. Thismeansthateachlinesegmentof L projectstoalinesegmentin R2satisfying thefollowingconditions: 3

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1.Theprojectionsoftwosegmentsintersectinatmostonepoint(i.e.,noprojectionsofthetwosegmentsoverlapinasubsegmentasdepictedinFigure1(a)). 2.Anytwosegmentsdonotintersectatanendpoint(seeFigure1(b)). 3.Nointersectionpointbelongstotheprojectionsofthreesegments(sothatthe situationdepictedinFigure1(c)doesnothappen). (a)(b) (c) Figure1.Restrictionsoftheprojectiononlinesegments 4

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Givensuchasituation,theimageof L in R2togetherwithoverandunderŽ informationatthecrossingsiscalleda linkdiagram of L andisdenotedby DL. A crossing isapointofintersectionoftheprojectionsoftwolinesegmentsof L TheoverandunderŽinformationreferstotherelativeheightsabove R2ofthetwo inverseimagesofacrossing.Thisinformati onisindicatedinpicturesbybreakingthe under-passingsegmentsasshowninFigure3 .Afterbreakingunder-passingsegments, theprojectionof L becomesadisjointunionof arcs .Thus,anarcisaconnected componentoftheprojectio nafterbreakingunder-passi ngsegments.Inpractice, wedrawwellroundedcurvesforknotdiagr ams(seeFigure2)insteadofpolygonal segments. Figure2.Awellroundedtrefoil Theprojectionmappreservestheorientationofalink.The co-orientation isa familyofvectorsin R2normaltothelinkdiagram,suchthatthepair(orientation, co-orientation)matchesthegivenorientat ion(right-handed,orcounterclockwise)of theplane.Atacrossing,ifthepairoftheorientationoftheover-arcandthatof theunder-arcmatchesthe(right-hand)orientationoftheplane,thenthecrossingis called positive ;otherwiseitis negative .InFigure3,thecrossingdepictedattheleft ispositiveandtheotherontherightisnegative. Denition1.2.4 [35]The sign ofacrossing ,denoted ( ),istakentobe1ifthe crossingispositiveand Š 1ifthecrossingisnegative,asillustratedinFigure3. 5

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( )=+1 ( )= Š 1 Figure3.Positiveandnegativecrossings Iftwolinks L1and L2areequivalent,thentheirrespectivediagrams DL1and DL2arerelatedbyasequenceof Reidemeistermoves andanorientation-preserving homeomorphismoftheplane.Inthiscase,thetwodiagrams DL1and DL2are equivalent .Therearethreesuchmovescalledthe“rst(TypeI),second(TypeII),and third(TypeIII)Reidemeistermove,respectively.ThethreeReidemeistermovesare depictedinFigure4(see,forexample,[30]).TheReidemeisterTypeImoveallows toputinortakeoutasmalltwistinthestring.Thesecondmoveisusedtoeither addtwocrossingsorremovetwocrossing slocally.TypeIIImoveallowstoslidea strandfromonesideofacrossingtotheother. TypeITypeII TypeIII Figure4.TheReidemeistermoves 6

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Denition1.2.5 [35]Apropertyofalink L ,oralinkdiagram DLrespectively,is invariant ifitremainsthesameforalllinks(resp.linkdiagrams)equivalentto L (resp. DL). 1.3Fox n -coloringofknotdiagrams ThemostelementaryknotinvariantisFoxs3-colorability.Sinceourinvariantsare generalizationsofthis,werevie witsde“nitioninthissection. Denition1.3.1 [20]Alinkis 3-colorable ifithasadiagram,suchthatwecan assigneither0,1or2(thesenumbersarecalled colors )toitsarcsinsuchawaythat thefollowingconditionsaresatis“ed: 1.eacharcisassignedasinglecolor, 2.atleasttwocolorsareused,and 3.ateachcrossing,eitherallarcshavethesamecolor,orallofthethreecolors meet.SeeFigure5.Inthe“gure, { a,b,c } = { 0 1 2 } ,i.e. a,b and c represent distinctnumbersfromtheset { 0 1 2 } b aa ac a Figure5.The3-colorabilitycondition Theorem1.3.2 [20]3-colorabilityisalinkinvariant. Proof. Thisisprovedbycheckingthatthe3-colorabilityremainsunchangedby theReidemeistermoves.Seesection1.3of[20]fordetails. Example1.3.3 Thetrefoil,asshowninFigure6,is3-colorable. 7

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2 0 1Figure6.Acoloredtrefoil Foranypositiveinteger n ,3-colorabilitycanbegeneralizedto n -colorabilityas follows.Let n beanaturalnumbergreaterthan2.Let a1,a2,...,akbethearcsof alinkdiagram.Assigntoeacharcaninteger i{ 0 1 ,...,n Š 1 } (called color ). Let qbethecolorassignedtotheover-arcand r,sbethecolorsassignedtothe twounder-arcsatacrossing.Then,itisrequiredthatthecondition r+ s 2 q(mod n )besatis“edateverycrossing.SeeFigure7. rqsFigure7.The n -colorabilitycondition Denition1.3.4 [35]Alinkissaidtobe Fox n -colorable ifforsomediagram DLof L ,colorscanbeassignedtothearcssatisfyingtheabovepropertiesusingatleast twodistinctcolors. Theorem1.3.5 [19] n -colorabilityisalinkinvariant. Inthenextchapterweintroduceanalgebra icstructurecalledquandlethatgeneralizes n -colorability. 8

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CHAPTER2 QUANDLES Thealgebraicstructure kei de“nedbyTakasaki[39]appearstobethe“rstoccurence ofaquandleintheliterature.Sincethen, similaralgebraicstructureshavebeen de“ned,butwereoftenmotivatedfromsymme trictransformations,ratherthanfrom knottheory.AdramaticchangeinthestudyofquandlesoccuredwhenJoyce[27], andatthesametimeMatveev[33],initiatedthestudyofquandlesinrelationtoknot theory.ThetermquandlewasinventedbyJoyce.Then,Brieskorn[3]introducedthe automorphicsets ,droppingtheidempotency( x x = x, x )condition,andpointedout manyoccurencesofthisstructure.Later,FennandRourke[15]calledthisstructure racks .Kauman[28]alsogaveadescriptionofasimilarstructurecalled crystal Furthermore,(co)homologytheoryforrackswasde“nedin[17],andfromthepoint ofviewofHopfalgebrasin[22].Amodi“edversionof(co)homologytheoryfor quandleswasdescribedin[8]forde“ningst ate-suminvariantsforknotsandknotted surfaces. Wegivede“nitionsandexamplesofquandlesinSections2.1and2.2.Then,in Section2.3weclassify4-elementquandles.Coloringsofknotdiagramsbyquandles arede“nedinSection2.4. 2.1Denitions Denition2.1.1 [8]A quandle X ,isasetwithabinaryoperation( a,b ) a b suchthat (I)Forany a X a a = a 9

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(II)Forany a,b X ,thereisaunique c X suchthat a = c b (III)Forany a,b,c X ,wehave( a b ) c =( a c ) ( b c ) .c ab a*bIIbc*b a b IIIabcabc ccb*c b*c a*c a*b (a*b)*c (a*c)*(b*c) a*aaa a aI a b Figure8.Reidemeistermovesandquandleconditions A rack isasetwithabinaryoperationthatsatis“es(II)and(III).Ifarackora quandleis“nite,thenthenumberofelementsin X iscalledthe order of X WecallAxiom(III)inthede“nitionofaquandlethe rackidentity .Notethatitis therightself-distributivelaw.Racksandquandleshavebeenstudiedin,forexample, [3,15,27,28,33]. ThethreeaxiomsforaquandlearisefromReidemeistermovesoftypeI,II,andIII, respectively[15,28].AtthetopleftofFigure8,acoloringruleisdepicted,whichwill bepreciselyde“nedinSection2.4.Underthiscoloringruleweobservethatthecolors atthebottomsegmentsof“guresI,II,andIII,matchbeforeandafterReidemeister movesI,II,andIII.Thus,Figure8showsthatReidemeistermovesoftypeI,II,and IIIcorrespondtothequandleaxiomsI,II,andIII,respectively.Quandlestructures havebeenfoundinareasotherthanknottheory,see[2,3]forexample. 10

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Denition2.1.2 Afunction f : X Y betweenquandlesorracksisa homomorphism if f ( a b )= f ( a ) f ( b )forany a,b X .Ahomomorphism f iscalledan isomorphism if f isone-to-oneandonto.Anisomorphism f : X X iscalledan automorphism [8,27]. Thereareseveralimmediateconsequen cesofthequandleandrackaxioms.Let X denoteaquandle.FromAxiom(II)ofDe“nition2.1.1,eachelement b X de“nesabijection S ( b ): X X with aS ( b )= a b (thefunctionisontheright). ThebijectionisaquandleautomorphismbyAxiom(III)ofDe“nition2.1.1.For a word w = b11 bnnwith b11,...,bnn X ; 1,...,n{ 1 } ,wede“ne a w = aS ( w )by aS ( b1)1 S ( bn)n,where S ( b )Š 1denotestheinversemappingof S ( b ). Theterminology S ( b )followsJoycespaper[27]and a w (= aw)followsFennand Rourke[15]. Denition2.1.3 [25]Let X denoteaquandle.Anautomorphismof X iscalled inner-automorphism of X ifitis S ( w )foraword w Denition2.1.4 [10]De“nearelationon X by a b if a ismappedto b byan inner-automorphismof X .Therelation isanequivalencerelation.The orbit of a X istheequivalenceclassof a ,whichisdenotedbyOrb( a ).Thesetofequivalence classesof X by isdenotedbyOrb( X ).WhenweregardOrb( X )asatrivialquandle (seeSection2.2),theprojectionmap : X Orb( X )isaquandlehomomorphism. InthiscaseOrb( X )iscalledthe orbitquandle of X If c b = a ,wewrite c = a b .Notethatif( X, )isaquandleorarack,thenso is( X, ).Itiscalledthe dualquandle of( X, )[8]. Asubsetofaquandlethatformsaquandlebyitselfiscalleda subquandle 11

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2.2Examplesofquandles € TheTrivialQuandle[15].Anyset T withtheoperation x y = x forany x,y X isaquandlecalleda trivial quandle.If T has n elements,theelements of T canberepresentedbythenumbers1 2 ,...,n ,and T isdenotedby Tn. € TheConjugationQuandle[15].Foranygroup G ,de“ne g h tobe hŠ 1gh for any g,h G .Then,( G, )de“nesthe conjugationquandle ,sometimeswritten by conj ( G ). € TheDihedralQuandle[15].Let D2 nbethedihedralgroupoforder2 n ,which maybetakentobethesymmetrygroupofaregular n -gon.Then, D2 nhasa presentation D2 n= x,y | x2=1= yn,xyx = yŠ 1 where x isare”ectionthrougha“xedvertexand y isarotationofaregular n gonthroughanangleof2 naboutitscenter.Theset Rnofallpossiblere”ections iswrittenas { ai= xyi| i =0 ,...,n Š 1 } ,anditisclosedunderconjugation. Weusethesubscriptsfrom Zninthefollowingcomputations.Theoperation byconjugationiscomputedas ai aj= aŠ 1 jaiaj= xyjxyixyj= xyjyŠ iyj= a2 j Š i. Therefore,wemayconsiderthedihedralquandleas Rn= { 0 1 2 ,...,n Š 1 } withtheoperation a b 2 b Š a (mod n ). € TheAlexanderQuandle[8].LetbetheringofLaurentpolynomials Z [ T,TŠ 1] inthevariable T .Any-module M hasthestructureofaquandlewiththe 12

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operation a b = Ta +(1 Š T ) b forany a,b M .Wecallsuchaquandlean Alexanderquandle .Inparticular,foranyLaurentpolynomial h ( T )suchthat thehighestandlowesttermsareinvertiblein Zn, Zn[ T,TŠ 1] / ( h ( T ))isa“nite quandle. If T = Š 1,thenthequandleoperationofa“niteAlexanderquandlebecomes a b 2 b Š a (mod n ),whichistheoperationfordihedralquandles.Thisvalue of T isachievedwhen h ( T )= T +1.If T =1,thenthequandleoperationbecomes a b = a ,whichistheoperationfortrivialquandles(ifapositiveinteger n isused insteadof0).Thisvalueoccursif h ( T )= T Š 1.Thus,bothdihedralandtrivial quandlesmaybeconsideredassp ecialcasesofAlexanderquandles. 2.3Classicationof 4 -elementquandles Inthissectionweusenewconstructionsofquandlestodescribe4-elementquandles, inadditiontoextensionsbycocycles(seeChapter4). Let X = { ( X, ): } beafamilyofracks,whereisanindexset.Arack W ,calledthe disjointunion of X ,isde“nedasfollows.Asaset, W = X .For x1 Xand x2 X,therackoperationisde“nedby x1 x2= x1x2if = x1if = Itischeckedthat W isarack(orquandle)if Xisforevery .Thisconstruction isfoundin[3,15]. Anotherconstructionisgivenasfollows.Let X0,X1betrivialquandles,andlet : X1A S ( X0)beamapintoanabeliansubgroup A ofthepermutationgroup ofelementsof X0.Theimageof neednotbeasubgroupof A .Denotetheimage 13

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by ( k )= k: X0 X0.Let X = X0 X1andde“neabinaryoperationon X by a b = a if a,b X0, or a X1b( a )if a X0,b X1Lemma2.3.1 Theaboveconstructed X isaquandlewith asitsoperation. Proof. Theconditions(I)and(II)ofDe“nition2.1.1areeasilycheckedcasebycase. If a X1,thenthebothsidesoftheself-distributivelaw(III)are a ,andthusthe lawissatis“ed.Axiom(III)isalsosatis“ed,ifall a,b,c arein X0.If a X0and b,c X1,then ( a c ) ( b c )=( a c ) b =( b c)( a )=( c b)( a )=( a b ) c, as A isanabeliangroup.If a,b X0and c X1,then ( a b ) c = c( a b )= c( a )= a c =( a c ) ( b c ) astheelement b c X0actstriviallyon X0.If a,c X0and b X1,then ( a b ) c = b( a ) c = b( a )= b( a c )=( a c ) b =( a c ) ( b c ) Theseexhaustallthecases. Example2.3.2 Let Vn= Tn Š 1 T1for X0= Tn Š 1,X1= T1inthepreviousconstruction,wheretherightactionof T1on Tn Š 1isapermutationand Tndenotesthe trivialquandleof n elements.Then Vnisaquandle.Inparticular,thereisaquandle V4offourelementsuptoisomorphism,where T1actsasa3-cycleon T3. Weusetheaboveconstructionsofquandlesasatooltodescribequandles,and asanapplicationweclassify4-elementquandles.Quandleswiththreeelementsare 14

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R4P3 T1Y4 y = y = y = x y abcdx y abc dx y abcd x = a aabbx = a ax = a aaaa b bbaab P3 bb bbbb c ddccc cc ddcc d ccddd dd ccdd Table1.Multiplicationtablesfor R4, P3 T1,and Y4classi“edin[15]: T3, R3,and P3where P3isthehomomorphicimageof R4witheven numbersidenti“edtoasingleelement. Let Y4bethequandlede“nedbythemultiplicationtabletotherightofTable1.Weremarkherethatusingtheextensionsofquandles E ( X,A, )arede“ned inSection4.1, R4isdescribedas E ( T2, Z2,0 1+ 1 0)andthe Y4isdescribedas E ( T2, Z2,0 1).SeeSection4.1formoredetailsonnotation. Proposition2.3.3 Any 4 -elementquandleisisomorphictoexactlyoneofthequandlesinthefollowinglist: T4, Z2[ T,TŠ 1] / ( T2+ T +1) V4, R3 T1, P3 T1, R4, Y4. Thusextensions,theirhomomorphicim ages,anddisjointunionsareexpectedto beeectivewaysofdescribingsmallquandles.TheproofofProposition2.3.3follows fromthefollowingsequenceoflemmas. Lemma2.3.4 Let X beaquandlewithfourelements.Ifthereare a,b X such that b a = b ,thenthereisatrivialsubquandle T2 X Proof. If a b = a ,then T2= { a,b } isatrivialsubquandle.Hence,assumethat a b = c ,where c isanelementof X distinctfrom a and b .Then, c a =( a b ) a = a ( b a )= a b = c (Case1) c b = a .Then, a c =( c b ) ( a b )=( c a ) b = c b = a ,sothat X wouldcontainasubquandle T2= { a,c } (ifthisinformationcompletestoforma quandle X ). 15

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(Case2) c b = d .Then, a c =( d b ) ( a b )=( d a ) b = d b = a ,so weobtain T2= { a,c } .Here, d a = d sincetheactionof a isapermutation,and wealreadyhave b a = b and c a = c .Similarly, d b = a sincewealreadyhave a b = c and c b = d Lemma2.3.5 Ifa4-elementquandle X doesnothaveatrivialsubquandle T2oftwo elements,then X isisomorphicto Z2[ T,TŠ 1] / ( T2+ T +1) Proof. Let a,b X bedistinctelements.ByLemma2.3.4,wemayassumethat b a = c ,where c X isdistinctfrom a and b .If c a = b ,then d a = d ,which implies T2 X byLemma2.3.4.Henceweassumethat c a = d .Then,wehave d a = b consideringthattheactionby a fromtherightisapermutation. (Case1)Suppose a b = a .ByLemma2.3.4thereis T2 X (Case2)Suppose a b = c .Wehave a c =( a a ) ( b a )=( a b ) a = c a = d .Then,byconsideringtheactionby b ,wehavethat c b canbeeither a or d .Similarly,byconsideringtheactionby c ,wehavethat b c canbeeither a or b Since c b =( b a ) b = b ( a b )= b c then c b = b c = a ,whichcontradicts b c =( d a ) ( b a )=( d b ) a = d a = b .Therefore,thischoicedoesnotyield aquandle. (Case3)Suppose a b = d .ByLemma2.3.4andactionsbyelements,thiscondition uniquelydeterminesaqu andleisomorphicto Z [ T,TŠ 1] / ( T2+ T +1).Speci“cally,the isomorphismisgivenby a 0, b 1, c T ,and d 1+ T Lemma2.3.6 Supposea 4 -elementquandle X hasasubquandleisomorphicto R3. Then X isisomorphicto R3 T1. Proof. Let X = { a,b,c,d } and R3= { a,b,c } .Sincetherightactionisinjective,we have d x = d for x = a,b,c .Also,forany x = a,b,c ,wehave x d = x ( d x )= ( x d ) x andthereisauniquesolution y for y x = x ,namely, y = x ,andweobtain x d = x .Theresultfollows. 16

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Lemma2.3.7 Supposeanon-trivialquandle X = { a,b,c,d } hastrivialsubquandles T2 = { a,b } and T2 = { c,d } .Then, X isisomorphictoeither P3 T1, R4or Y4. Proof. Iftherightactionof { a,b } on { c,d } istrivial,then c a = c,d a = d,c b = c,d b = d .Then,therearearetwopossibilities,either a c = a or a c = b .When a c = a and a d = a ,then X isisomorphicto T4.Otherwise,when a c = a and a d = b wehavethat X isisomorphicto P3 T1.If a c = b and a d = a ,then X isisomorphicto P3 T1.Otherwise,for a c = b and a d = b X isisomorphicto Y4. Supposetherightactionof { a,b } on { c,d } isnottrivial.Therearethreesuch cases:(Case1) c a = c,d a = d,c b = d,d b = c ,inwhichcase, a d = ( a b ) ( c b )=( a c ) b .If a c = a ,then a d = a b = a ,and P3 T1results.If a c = b ,then a d = b b = b ,andthequandle X hasthefollowingmultiplication table: y = x y abcd x = a aabb b bbaa c cdcc d dcdd However,( c a ) c = c and c ( a c )= d ,sothatthisisnotaquandle. (Case2) c a = d,d a = c,c b = c,d b = d ,whichissimilartothecaseabove. (Case3) c a = d,d a = c,c b = d,d b = c ,inwhichcase, a d = a ( c a )= ( a c ) a .If a c = a ,then a d = a a = a ,andthequandle Y4appears.If a c = b then a d = b a = b ,inwhichcase X = R4. Lemma2.3.8 Suppose X = { a,b,c,d } hasatrivialsubquandle T2= { a,b } ,and { c,d } doesnotformatrivialquandle.Then X isisomorphicto R3 T1or V4. 17

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Proof. Therearethreecasesfortherightactionsof { a,b } on c .(Case1) c a = c,c b = c .Itfollowsthat d a = d,d b = d .Ifthereisasubquandle T3,then X is isomorphicto V4.Therefore,assumethatthereisnosubquandle T3.Since a and b playsymmetricroles,therearetwocases: d c = a ,and d c = d .If d c = a ,then a c = b and b c = d (otherwiseitfallsintothenextcase,byswitching b and d ). Then, a d =( d c ) ( b c )=( d b ) c = d c = a and b d =( a c ) ( b c )=( a b ) c = a c = b, thus X hasasubquandle T3.If d c = d ,then a c = b and b c = a ,sinceotherwise ithasasubquandle T3.Notethatinthiscase a d =( b c ) d =( b d ) ( c d ),and b d =( a c ) d =( a d ) ( c d ).Hence, c d transposes a d and b d byright action.Thisimpliesthat c d = c ,then { c,d } istrivial. (Case2) c a = c and c b = d .If d c = a ,then c d =( d c ) b = a b = a and a c =( c d ) c = c ( d c )= c a = c ,whichcontradictstheactionof c onthe remainingelementsbypermutation.Hence,thischoicedoesnotyieldaquandle.If d c = b ,then c d =( d c ) b = b b d =( d c ) d = d ( c d )= d b = c ,and b c =( c d ) c = c ( d c )= c b = d ,and { b,c,d } formasubquandleisomorphic to R3. Suppose d c = d .Then, c d =( d c ) b = d b = c ,and { c,d } form T2.We observethatthecase c a = d and c b = c issimilar. (Case3) c a = d,c b = d .Assume d c = d .Then, c d =( d c ) a = d a = c thus { c,d } isatrivialquandle.Hence,either d c = a or b .Onecaseissimilarto theother,soassume d c = a .Itfollowsthat c d =( d c ) a = a ,aswellas a c =( c d ) c = c ( d c )= c a = d ,and a d =( d c ) d = d ( c d )= d a = c 18

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Hence,thereisasubquandle R3= { a,c,d } .Infact,since c b = d ,thiscase contradictsLemma2.3.6anddoesnotformaquandle. Itiseasytoseethatthequandleslistedi nProposition2.3.3arenotisomorphic toeachother,bylookingatsubquandlesororbits.Thisisprovedinthefollowing lemma. Lemma2.3.9 Thequandles T4, Z2[ T,TŠ 1] / ( T2+ T +1) V4, R3 T1, P3 T1, R4, Y4arepairwisenon-isomorphic. Proof. Thequandleslistedabove,exceptfor T4,arenottrivialquandles.Theonly propersubquandleof Z2[ T,TŠ 1] / ( T2+ T +1)is T1,whiletheremainingquandleshave propersubquandlesofhigherorder.Thequandle V4istheonlyquandlefromthelist thathasapropersubquandleisomorphictoatrivialquandle T3.ByLemma2.3.8, ifaquandlecontainsasubquandleisomorphicto T3,thenthatquandlemustbe V4. Thelargestsubquandlethatisiso morphictoatrivialquandleis T2fortheremaining quandles, R3 T1, P3 T1, Y4,and R4.ByLemma2.3.6, R3 T1istheonlyquandle thatcontainsasubquandleisomorphicto R3. Thequandle P3 T1hasoneelementthatacts(fromtheright)triviallyonthe remainingelements.Thiscanbeseenfromamultiplicationtablefor P3 T1,shown inthemiddleofTable1,where d actstriviallyon a,b,c and d .However, R4hasno suchelement,asshowninthemultiplicatontablecorrespondingto R4inTable1. Amultiplicationtableforthequandle Y4isgiveninTable1.Therearetwo elements, c and d ,thatacttriviallyontheotherthreeelements.Theseresultthe threeelementsubquandlesshowninTable2,whichdierfromthequandle P3,also showninTable2.Hence, Y4isnotisomorphicto P3 T1. ByLemma2.3.7, R4and Y4appearwhenwehavetwotrivialsubquandlesof ordertwo.Thequandle Y4hasanelement a suchthatalltheelementsacton a trivially.Thequandle R4,however,doesnothavesuchanelement.Therefore,the twoextensionsarenotisomorphic. 19

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T1= { c } T1= { d } P3 y = y = y = x y abdx y abcx y 013 x = a aaax = a aaax =0 000 b bbbb bbb 1 311 d ccdc ddc 3 133 Table2.Comparisonof3-elementquandlesof Y4with P3Recently,Gra na[21]classi“edindecomposableracks(anextensionofarackwith noproperquotients)oforder p2,where p isprime.Nelson[36]obtainedaprocedure forclassifying“niteAlexanderqua ndlesintermsoftheirsubmodules. Animportantprobleminknottheoryistodeterminewhichknotsareequivalent andwhicharenot.Onewayofapproachingthisproblemisbycoloringknotdiagrams. InthenextsectionweseethatFoxs n -coloringgeneralizestocoloringsbyquandles. 2.4Coloringsofknotdiagramsbyquandles Themotivationforstudyingquandlespa rtlyarisesfromknotandlinkdiagrams. Consideranorientedknotdiagram,withco-orientaiongivenbytherighthandrule. Let X beaquandle.Itispossibletolabeleacharcoftheknotdiagrambyaquandle elementasfollows. Denition2.4.1 [28,35]A coloring ofanorientedclassicallinkdiagramisafunction C : R X ,where X isa“xedquandleand R isthesetofover-arcsinthediagram, satisfyingtheconditiondepictedinFigure9.Inthe“gure,acrossingwithover-arc, ,hascolor C ( )= b X .Theunder-arcsarecalled and fromtoptobottom; thenormal(co-orientation)oftheover-arc pointsfrom to .Then,itisrequired thatif C ( )= a and C ( )= c ,then c = a b Thequandleelement C ( r )assignedtoanarc r byacoloring C iscalleda color of thearc.Notethatlocallythecolorsdon otdependontheorientationoftheunderarc.Thisde“nitionofcoloringsonknotdiagramshasbeenknown,see[15,19]for 20

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C ( )= b C ( )= c = a b C ( )= a Figure9.Quandlerelationatacrossing example.Thesetofcoloringsofaknotdiagram K byaquandle X isdenotedby ColX( K ).Thecardinalityofallsuchcoloringsisdenotedby | ColX( K ) | .Henceforth, allthequandlesthatareusedtocolordiagramswillbe“nite. Example2.4.2 Let X be R3(thedihedralquandleoforder3)withquandleoperation a b =2 b Š a ,where a,b,c X .Then,thetrefoilKiscoloredby R3asdepicted inFigure10.Itisseenthat | ColX( K ) | =9.2 2*0 = 1 0*1 = 2 1*2 = 0 1 0 Figure10.Atrefoilcoloredbythequandle R3Proposition2.4.3 [8]Thenumberofcolorings | ColX( K ) | isaknotinvariant. Considerthedihedralquandleof n elements, Rn.Atacrossingwehavethecolors C ( r)= a, C ( q)= b and C ( s)= a b .If a = b ,then a b = a a = a .Otherwise, a b 2 b Š a (mod n ).Weseethatthequandleoperationisequivalenttothe requirementofthe n -colorabilityconditionde“nedinDe“nition1.3.4.Therefore,an n -coloringisaquandlecoloringofalinkby Rn.The n -colorabilityisequivalentto | ColX( K ) | > | X | .Theclassicalresultthataknotisnon-trivially(Fox) n -colorable 21

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(for n prime)if n | ( Š 1),where( T )denotestheAlexanderpolynomial,hasbeen generalizedbyInoue[24]tothefollowing: Theorem2.4.4 [24]Let( i ) K( T )denotethegreatestcommondivisorofall( n Š i Š 1) minordeterminantsofthepresentation matrixfortheknotmoduleobtainedvia theFoxcalculus[29].Let p beaprimenumberand J anidealoftheringp= Zp[ T,TŠ 1].Foreach i 0,put ei( T )=( i ) K( T ) / ( i +1) K( T ) Then,thenumberof coloringsbytheAlexanderquandlep/J isequaltothecardinalityofthemodule p/J n Š 2 i =0{ p/ ( ei( T ) ,J ) } Classicalknotshavefunda mentalquandlesthatarede“nedviageneratorsand relations.Thede“nitionsofthefundamentalquandlearefoundin[15,27,28,33], forexample.Herewegiveabriefdescription. A presentation S | R ofarackoraquandleisde“nedinasimilarwayasfor groupsasfollows[15,27,28].Thefreerack FR ( S )isasaset S F ( S ),where F ( S ) isthefreegroupon S .Therackoperationisde“nedby( a,w ) ( b,z )=( a,wzŠ 1bz ) Thesetofrelations R isgivenanditconsistsofidentitiesoftheform x = y .De“nea congruencerelation onarack Y tobeanequivalencerelationsuchthatif a b Y then a c b c and c a c b ,forany c Y .Let on FR ( S )bethesmallest congruencecontaining R ,thatis, isthesmallestcongruencesuchthatif x = y is in R ,then x y .Then,therackwithgivenpresentationisde“nedby X = S | R = FR ( S ) / Forapresentationofaquandlewerequire a a a ,forany a FR ( S ). Thefundamentalquandleisde“nedinasimilarwaytothefundamentalgroup, asfollows.Thegenerators, x1,...,xm,areassignedtoarcsofagivenknotorlink diagram.ArelationisassignedtoeachcrossingasdepictedinFigure9.Speci“cally, if xiisthegeneratorassignedtotheunder-arcawayfromwhichthenormalofthe 22

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over-arcpoints, xkisassignedtotheotherunder-arc,and xjisassignedtotheoverarc,thentherelation rh: xi xj= xkisassignedtoobtainthesetofrelations r1,...,rnfromallthecrossings(seeFigure11). xjxkxirh: xi xj= xkFigure11.Wirtingerrelationofthefundamentalquandle Thequandle Q ( K )de“nedbythethusobtainedpresentation x1,...,xm| r1,...,rn iscalledthe fundamentalquandle ofaclassicalknot K ,orsimplythe quandleof K NotethesimilarityoftheWirtingerpres entationtothefundamentalgroup[30], wheretherelation xi xj= xkcorrespondsto xŠ 1 jxixj= xkinthefundamentalgroup. Example2.4.5 Fromastandarddiagramofatrefoil K asdepictedinFigure6,we obtainapresentationforthefundamentalquandleofthetrefoil Q ( K )= x1,x2,x3| x3 x2= x1,x2 x1= x3,x1 x3= x2 Acoloringofaclassicalknotdiagrambyaquandle X givesrisetoaquandle homomorphismfromthefundame ntalquandletothequandle X [24].Knotdiagrams coloredbyquandlescanbeusedtostudyqu andlehomologygroups.Thisviewpoint wasdevelopedin[16,18,23]forrackhomologyandhomotopy,andgeneralizedto quandlehomologyin[12].Quandlehomomorphismsandvirtualknotsareappliedto thishomologytheory[10].State-suminva riantsusingquandlecocyclesasBoltzmann weightsarede“ned[8]andcomputedforimp ortantfamiliesofclassicalknotsand 23

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knottedsurfaces[9].Theinvariantswerea ppliedtostudyingknots,forexample,in detectingnon-invertibleknottedsurfaces[8]. Thenextchapterdiscusseshomologyandcohomologytheoriesofquandles,and cocycleknotinvariants. 24

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CHAPTER3 COHOMOLOGYTHEORYOFQUANDLESANDCOCYCLEKNOT INVARIANTS Inthischapter,following[10]and[25],we“rstde“nethehomologyandcohomology theoriesforquandlesneededtounderstand thiswork.Then,motivatedbythemain problemofknottheoryofdistinguishingd ierentknots,weconsidernewlinkinvariants.Thenewinvariantsarede“nedusingcoloringsoflinkdiagramsbyquandles andquandlecocycles. 3.1Homologyandcohomologyofquandles Originally,rackhomologyandhomotopytheorieswerede“nedandstudiedin[16],and amodi“cationtoaquandlehomologytheorywasgivenin[8]tode“neaknotinvariant inastate-sumform.Themostgeneralformofthequandlehomologyknowntodate isgivenin[2].Thecohomologytheoryhasf oundapplicationstotheclassi“cationof Nicholsalgebras[2].Computationsarefoundin[9,10,14,31,34],forexample. For n> 0,let CR n( X )bethefreeabeliangroupgeneratedby n -tuples( x1,...,xn) ofelementsofaquandle X .If n 0,let CR n( X )=0.De“neahomomorphism n: CR n( X ) CR n Š 1( X )by n( x1,x2,...,xn)=ni =2( Š 1)i[( x1,x2,...,xi Š 1,xi +1,...,xn) Š ( x1 xi,x2 xi,...,xi Š 1 xi,xi +1,...,xn)] for n 2and n=0for n 1.Directcalculationsshowthat =0.Then, CR ( X )= { CR n( X ) ,n} isachaincomplex. 25

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Let CD n( X )bethesubsetof CR n( X )generatedby n -tuples( x1,...,xn)with xi= xi +1forsome i { 1 ,...,n Š 1 } if n 2;otherwiselet CD n( X )=0.If X isaquandle, then n( CD n( X )) CD n Š 1( X )and CD ( X )= { CD n( X ) ,n} isasub-complexof CR ( X ). Put CQ n( X )= CR n( X ) /CD n( X )and CQ ( X )= { CQ n( X ) n} ,where nistheinduced homomorphism.Henceforth,all boundarymapswillbedenotedby n. Foranabeliangroup G ,de“nethechainandcochaincomplexes CW ( X ; G )= CW ( X ) G, = id; C W( X ; G )=Hom( CW ( X ) ,G ) =Hom( id) intheusualway,whereW=D,R,Q. Denition3.1.1 [8]The n th quandlehomologygroup andthe n th quandlecohomologygroup ofaquandle X withcoecientgroup G are HQ n( X ; G )= Hn( CQ ( X ; G )) ,Hn Q( X ; G )= Hn( C Q( X ; G )) Thecycleandboundarygroupsaredenotedby ZW n( X ; G )and BW n( X ; G ),sothat HW n( X ; G )= ZW n( X ; G ) /BW n( X ; G ),whereWisoneofD,R,orQ.Thecocycleand coboundarygroupsaredenotedby Zn W( X ; G )and Bn W( X ; G ),respectively,sothat Hn W( X ; G )= Zn W( X ; G ) /Bn W( X ; G ) Thecoecientgroup G isomittedif G = Z Inthefollowingsectionswediscusscocycleknotinvariantsfromtheviewpointof coloringknotdiagramsbyquandles. 26

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3.2Cocycleknotinvariants Thenotionofstate-sumoriginatedfromstatisticalmechanics.Fromthemathematicalpointofview,thestate-suminvariantshavebeenstudiedinrelationtotheJones polynomialandgeneralizations(see,forexample,[35]).State-suminvariantsusing quandlecocyclesasBoltzmannweightswerede“nedin[8]andcomputedforimportantfamiliesofclassicalknotsandknotteds urfacesin[9].Theinvariantswereapplied tostudyingknots,forexample,indetectin gnon-invertibleknottedsurfaces[8].Here wedescribesuchinvariants. InFigure12,thetwopossibleorientedandco-orientedcrossingsaredepicted.On theleftthecrossingispositiveandontherightisnegative.Let denoteacrossing and C denoteacoloring.Let r betheover-arcat ,and r1, r2beunder-arcssuchthat thenormalto r pointsfrom r1to r2.Let x = C ( r1)and y = C ( r ).Pickaquandle2cocycle Z2( X ; G ).Then,de“ne B ( C )= ( x,y ) ( )tobethe Boltzmannweight where ( )=1or Š 1,if ispositiveornegativecrossing,respectively. x y xyy x ( x,y )Š 1 ( x,y ) x y Figure12.Weightsforpositiveandnegativecrossings Denition3.2.1 [8]The state-sum ,or partitionfunction ,isde“nedby ( K )= CB ( C ) Theproductistakenoverallcrossingsofthegivendiagram,andthesumistaken overallpossiblecolorings. 27

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Notethatthestate-sumdependsonthechoiceofthe2-cocycle .Thevaluesof thestate-sumaretakentobeinthegroupring Z [ G ]where G isthecoecient group(writtenmultiplicatively).Thisisproved[8]tobeaknotinvariant,calledthe (quandle)cocycleinvariant ( x, y ) ( p, q) ( q, r ) (p*q, r )( p, r ) ( q, r ) ( p*r, q*r ) p xy qr x*y q pr p*q q*r(p*q)*r(p*r)*(q*r) p*r q*r Figure13.TypeIIImoveandthequandleidentity Figure13showstheinvarianceofthestate-sumundertheReidemeistertypeIII move.Theproductsofcocycles,equatedbeforeandafterthemove,isthe2-cocycle condition ( p,q ) ( p q,r )= ( p,r ) ( p r,q r ) Example3.2.2 Considerthediagramofthe(4,2)-toruslink K asdepictedinFigure14.Let X = R4(thedihedralquandleoforder4),andlet G = Z2bethe coecientgroupwithgenerator t .De“nethecharacteristicfunctionby x( y )= t if x = y 1if x = y 28

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( ) i b k j a l b a i b k b k j a j a l b l b a i b k k b a j b l i a ( ) ( ) ( ) aFigure14.The(4,2)-toruslink where x,y denotepairs( a,b )ofquandleelements.Let beafunction : { 0 1 2 3 } { 0 1 2 3 }{ 1 ,t 1,t 2,... } givenby = 0 1+ 0 3.Itisknownthat satis“es the2-cocyclecondition[8].Withthiscocycle,thecolor ai=0 ,bk=1inFigure14 contributes t tothestate-sum.Notethatthechoiceofcolorsfor aiand bkuniquely determinesacoloringof L .Thereare16coloringsintotal.Thestate-suminvariant, asde“nedinDe“nition3.2.1,iscomputedbyconsideringallpossiblecoloringsas follows: CB ( C )= C ( ai,bk) ( bk,aj) ( aj,b) ( b,ai) = Ctq,q Z2=8+8 t. Therefore,we“ndthatthestate-sumis( T (4 2))=8+8 t 3.3Variationsofcocycleknotinvariants Motivatedbystate-suminvariantsofknotdiagrams,wegeneralizecocycleknotinvariantstoinvariantsofclassicallinkscomponent-wise. 29

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Denition3.3.1 Let L = K1 Krbealinkdiagramandlet Ti, i =1 ,...,r denotethesetofcrossingsatwhichtheunder-arcsbelongtothecomponent Ki. De“nethestate-sumi( L )= C TiB ( C )foreach i =1 ,...,r .Thevector ( L )=(i( L ))r i =1ofthestate-suminvariantsiscalledthe component-wise (quandle) cocycleinvariantof L Itwasobserved[5]that ( L )=( C TiB ( C ))r i =1isalinkinvariant,strictly strongerthanthesinglestate-sum. Example3.3.2 Wecaclulatethecomponent-wisecocycleinvariant,asgiveninDe“nition3.3.1,forthe(4,2)-toruslinkshowninFigure14.Thelinkhastwocomponents; let K1bethecomponentontheleftofthe“gureand K2thecomponentontheright. Let 1,2,3,4bethefourcrossingsfromtoptobottom,and Z2 Q( R4; Z2)be givenby 0 1+ 0 3.Let t bethegeneratorofthegroup Z .Then, T1= { 1,3} and T2= { 2,4} Forthe“rstcomponent, K1,wehave 1( L )= C T1B ( C ) = C ( ai,bk) ( aj,b) =12+4 t. Similarlyforthesecondcomponent K2weget2( L )= C ( bk,aj) ( b,ai)=12+4 t Therefore,thecocycleinvariantofthe(4,2)-toruslinkisgivenby ( T (4 2))=(1( T (4 2)) 2( T (4 2)))=(12+4 t, 12+4 t ) Lopes[32]observedthatthefamily { B ( C ) }C ColX( K ),whereColX( K )denotesthesetofcoloringsoftheknot K ,isaknotinvariantwithouttakingsumma30

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tion.Inparticular,in“nitequandlescanb eusedforcoloringinthiscase.Moreover, hede“nedfor r -componentlinksthevectorversion( { TiB ( C ) }C)r i =1. WecombinetheabovevariationstogetherwiththeonegivenbyDe“nition3.3.1 tode“nethefollowinggeneralizedcocycleinvariant. Denition3.3.3 [7]Let X beaquandle, Z2 Q( X ; A ),where A isanabelian group,let C beacoloringof L by X ,and B ( C )theBoltzmannweightatacrossing .Let L = K1 Krbealinkand Ti, i =1 ,...,r ,bethesetofcrossingsof L suchthattheunder-arcsbelongto Ki.De“ne ( L )= T1B ( C ) ,..., TrB ( C ) C ColX( L ), whereColX( L )denotesthesetofcolorings,i.e.ColX( L )= {C} ThisversionofafamilyofvectorsispotentiallystrongerthanLopessversionofa vectoroffamilies.Forexample,thetwodistinctfamiliesofvectors { (1 ,t ) ( t, 1) } and { (1 1) ( t,t ) } giverisetothesamevectoroffamilies( { 1 ,t } { 1 ,t } ).Asanexample, weevaluatetheinvariantforthe(4 2)-toruslink. Example3.3.4 WeapplyDe“nition3.3.3toFigure14ofthe L =(4 2)-toruslink. Thelink L = K1 K2,where K1isthecomponentontheleftofthe“gure,and K2isthecomponentontheright.Let X bethequandle X = R4andthecocycle = 0 1+ 0 3,where Z2 Q( R4; Z2),with Z2= { 1 ,t } .Thegeneralizedcocycle invariantiscalculatedtobe ( L )= T1B ( C ) T2B ( C ) C ColX( L )= { ( ( ai,bk) ( aj,b) ( bk,aj) ( b,ai)) }ColX( L )= { (1 1) ,..., (1 1) 8copies, (1 ,t ) ,..., (1 ,t ) 4copies, ( t, 1) ,..., ( t, 1) 4copies} 31

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CHAPTER4 EXTENSIONSOFQUAN DLESBY2-COCYCLES Inthischapterwediscussabelianextensionsofquandles.Constructionsofextensionsofquandlesusingcocyclesweregivenin[11],whicharesimilartoextensionsof groupsusinggroupcocycles[4].Wedevelo pmethodsofconstructingcocyclesfrom extensions.Thisistheoppositedirectionof[11]whereanextensionwasconstructed froma2-cocycle. 4.1Abelianextensions Foraquandle X ,anabeliangroup A ,anda2-cocycle Z2 Q( X ; A ),the abelian extension E = E ( X,A, )wasde“nedin[11]astheset A X ,withthequandle operationde“nedby( a1,x1) ( a2,x2)=( a1 ( x1,x2) ,x1 x2).Here,theabelian groupoperationof A inthe“rstfactorisdenotedbymultiplicativenotation.The followinglemmaistheconverseofthefactprovedin[11]that E ( X,A, )isaquandle. Lemma4.1.1 Let X E benitequandles,and A beaniteabeliangroupwritten multiplicatively.Supposethereexistsabijection f : E A X withthefollowing property.Thereexistsafunction : X X A suchthatforany ei E ( i =1 2 ), if f ( ei)=( ai,xi) ,then f ( e1 e2)=( a1 ( x1,x2) ,x1 x2) .Then, Z2 Q( X ; A ) Proof. Forany x X and a A ,thereis e E suchthat f ( e )=( a,x ),and ( a,x )= f ( e )= f ( e e )=( a ( x,x ) ,x ) sothatwehave ( x,x )=1forany x X 32

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Byidentifying A X with E by f ,thequandleoperation on A X isde“ned forany( ai,xi), i =1 2,by ( a1,x1) ( a2,x2)=( a1 ( x1,x2) ,x1 x2) Since A X isquandleisomorphicto E underthis ,wehave [( a1,x1) ( a2,x2)] ( a3,x3) =( a1 ( x1,x2) ,x1 x2) ( a3,x3) =( a1 ( x1,x2) ( x1 x2,x3) ( x1 x2) x3) and [( a1,x1) ( a3,x3)] [( a2,x2) ( a3,x3)] =( a1 ( x1,x3) ,x1 x3) ( a2 ( x2,x3) ,x2 x3) =( a1 ( x1,x3) ( x1 x3,x2 x3) ( x1 x3) ( x2 x3)) areequalforany( ai,xi), i =1 2 3.Hence, satis“esthe2-cocyclecondition. ThenLemma4.1.1impliesthat,underthesameassumptions,wehave E = E ( X,A, ),where Z2 Q( X ; A ).Nextweidentifysuchexamples. Theorem4.1.2 Foranypositiveintegers q and m Um +1= Zqm +1[ T,TŠ 1] / ( T Š 1+ q ) isanabelianextension E = E ( Zqm[ T,TŠ 1] / ( T Š 1+ q ) Zq, ) of X = Um= Zqm[ T,TŠ 1] / ( T Š 1+ q ) forsomecocycle Z2 Q( X ; Zq) Proof. Representtheelementsof Zqm +1by { 0 1 ,...,qm +1Š 1 } andexpressthem intheir q -aryexpansion: A = Amqm+ + A1q + A0 Zqm +1, 33

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where0 Aj
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s ( X Y ).Hence, s ( X ) s ( Y ) Š s ( X Y )isdivisibleby qm.Then,wehave f ( A B )=( Am+ ( A, B ) A B ) Therefore, f yieldsanisomorphism Zqm +1[ T,TŠ 1] / ( T Š 1+ q ) E ( Zqm[ T,TŠ 1] / ( T Š 1+ q ) Zq, ) Wecanalsomakethefollowingobservation.In Zqm,wehavethat A B =m Š 1j =0( AjŠ Aj Š 1+ Bj Š 1) qj. Moreover,theright-handsideofthisequa lityisauniquelydeterminedintegerfora given A,B Zqm +1.Ifthisintegerispositive,then A B canberewrittenasa q -ary expansionwith AmŠ Am Š 1+ Bm Š 1astheleadingcoecient,andwehave f ( A B )=( AmŠ Am Š 1+ Bm Š 1,m Š 1j =0( AjŠ Aj Š 1+ Bj Š 1) qj) Zq X. Ifthisintegerisnegative,thenrewrite A B intermsof q -aryexpansionwithpositive coecients,andapply f togetthat f ( A B )=( AmŠ Am Š 1+ Bm Š 1Š 1 ,m Š 1j =0( AjŠ Aj Š 1+ Bj Š 1) qj) Zq X. Thus,de“ne : Zqm Zqm{ 0 Š 1 } by ( A, B )= 0if m Š 1 j =0( AjŠ Aj Š 1+ Bj Š 1) qj 0 Š 1if m Š 1 j =0( AjŠ Aj Š 1+ Bj Š 1) qj< 0 35

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Then,werewrite f ( A B )as f ( A B )=( Am+ ( A, B ) A B ) where ( A, B )= Bm Š 1+ ( A, B ).Hence, f yieldsanisomorphism f : Zqm +1[ T,TŠ 1] / ( T Š 1+ q ) E ( Zqm[ T,TŠ 1] / ( T Š 1+ q ) Zq, ) Theorem4.1.3 Foranypositiveinteger q and m ,thequandle Wm +1= Zq[ T,TŠ 1] / (1 Š T )m +1isanabelianextensionof X = Wm= Zq[ T,TŠ 1] / (1 Š T )mover Zq: E = E ( X, Zq, ) ,forsome Z2 Q( X ; Zq) Proof. Representelementsof E by A = Am(1 Š T )m+ + A1(1 Š T )+ A0,where Aj Zq, j =0 ,...,m .De“ne f : E Zq X by f ( A )=( Am(mod q ) A (mod(1 Š T )m)) where A = m Š 1 j =0Aj(1 Š T )j. Then,for A,B Wm +1,thequandleoperationis computedtobe A B = TA +(1 Š T ) B =[1 Š (1 Š T )]( Am(1 Š T )m+ + A1(1 Š T )+ A0) +(1 Š T )( Bm(1 Š T )m+ + B1(1 Š T )+ B0) =( AmŠ Am Š 1+ Bm Š 1)(1 Š T )m+( Am Š 1Š Am Š 2+ Bm Š 2)(1 Š T )m Š 1+ +( A1Š A0+ B0)(1 Š T )+ A0=( AmŠ Am Š 1+ Bm Š 1)(1 Š T )m+m Š 1j =0( AjŠ Aj Š 1+ Bj Š 1)(1 Š T )j, 36

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where AŠ 1,BŠ 1areunderstoodtobezerosinthelastsummation,andthecoecients arein Zq.Notethatin Zq[ T,TŠ 1] / (1 Š T )mthequandleoperationgives A B =m Š 1j =0( AjŠ Aj Š 1+ Bj Š 1)(1 Š T )j. Therefore,wehave f ( A B )=( AmŠ Am Š 1+ Bm Š 1,m Š 1j =0( AjŠ Aj Š 1+ Bj Š 1)(1 Š T )j) Zq X. Then,wecanwrite f ( A B )=( Am+ ( A, B ) A B ) where ( A, B )= Bm Š 1Š Am Š 1.Hence f yieldsanisomorphism Zq[ T,TŠ 1] / (1 Š T )m +1 E ( Zq[ T,TŠ 1] / (1 Š T )m, Zq, ) Thecocycle hasasimilardescriptiontotheoneinTheorem4.1.2.Let s : Zq[ T,TŠ 1] / (1 Š T )m Zq[ T,TŠ 1] / (1 Š T )m +1beaset-theoreticsectionde“nedby s m Š 1j =0Aj(1 Š T )jmod(1 Š T )m =m Š 1j =0Aj(1 Š T )jmod(1 Š T )m +1. Then,wehave s ( X ) s ( Y )= s ( X Y )forany X,Y Zq[ T,TŠ 1] / (1 Š T )m.Hence, [ s ( X ) s ( Y ) Š s ( X Y )]isdivisibleby(1 Š T )m,andweget ( A, B )=[ s ( A ) s ( B ) Š s ( A B )] / (1 Š T )m Zq. 37

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Example4.1.4 1.Considerthecase q =2, m =2inTheorem4.1.2.Inthiscase Z4[ T,TŠ 1] / ( T +1)= R4, and Z8[ T,TŠ 1] / ( T +1)= R8= E ( R4, Z2, ) forsome Z2 Q( R4; Z2).Inordertoconstructthecocycle,weusethebijection f : R8 Z2 R4de“nedby f (0)=(0 0) ,f (1)=(0 1) ,f (2)=(0 2) ,f (3)=(0 3) f (4)=(1 0) ,f (5)=(1 1) ,f (6)=(1 2) ,f (7)=(1 3) ByLemma4.1.1,if f ( ei)=( ai,xi),then f ( e1 e2)=( a1+ ( x1,x2) ,x1 x2),using additivenotation,forany e1,e2 R8. Let e1=0 ,e2=1.Then f ( e1 e2)=(0+ (0 1) 0 1)=(0+ (0 1) 2).On theotherhand, f ( e1 e2)= f (0 1)= f (2)=(0 2).Byequatingthecorresponding partsinthelasttworelations,weobservethat (0 1)=0. Nowchoose e1=0and e2=2.Bysimilarcalculationswegetthat f ( e1 e2)= (0+ (0 2) 0 2)=(0+ (0 2) 0),and f ( e1 e2)= f (0 2)= f (4)=(1 0).Note thatthe“rstfactorsinthetworelationsdierby1.Thismustbethecontribution of (0 2).Therefore,thecharacteristicfunction 0 2appearsinthecocycle ,where a,b( x,y )= 1if( x,y )=( a,b ) 0if( x,y ) =( a,b ) denotesthecharacteristicfunction. 38

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Bycarryingoutsimilarcomputationsfora llpairs,weobtainanexplicitformula forthiscocycle : = 0 2+ 0 3+ 1 0+ 1 3+ 2 0+ 2 3+ 3 0+ 3 1. Similarcomputationsyieldthefollowingformulas. 2.Incase m =1and q =3,thecocycleconstructedisoftheform = 0 1+ 1 2+ 2 0+2( 0 2+ 1 0+ 2 1) 3.Incase m =2and q =3,thecocycleis = 0 3+ 0 4+ 0 5+2 0 6+2 0 7+2 0 8+2 1 0+ 1 4+ 1 5+ 1 6+2 1 7+2 1 8+2 2 0+2 2 1+ 2 5+ 2 6+ 2 7+2 2 8+2 3 0+2 3 1+ 3 5+ 3 6+ 3 7+2 3 8+2 4 0+2 4 1+2 4 2+ 4 6+ 4 7+ 4 8+ 5 0+2 5 1+2 5 2+2 5 3+ 5 7+ 5 8+ 6 0+2 6 1+2 6 2+2 6 3+ 6 7+ 6 8+ 7 0+ 7 1+2 7 2+2 7 3+2 7 4+ 7 8+ 8 0+ 8 1+ 8 2+2 8 3+2 8 4+2 8 5. 4.Considerthecase q =2and m =2inTheorem4.1.3.Thequandle Z2[ T,TŠ 1] / (1 Š T )2isisomorphicto R4bythecorrespondence0 0(1 Š T )+0,1 0(1 Š T )+1, 2 1(1 Š T )+0,and3 1(1 Š T )+1.Thisisaspecialcaseoftheisomorphism Zn[ T,TŠ 1] / (1 Š T )2 = Zn2[ T,TŠ 1] / ( T Š ( kn +1))ifgcd( n,k )=1 39

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givenin[31].Then,thequandle Z2[ T,TŠ 1] / (1 Š T )3isanabelianextension E ( R4; Z2,) forsome Z2 Q( R4; Z2).Moreover,thecocycle ( A, B )= B1Š A1is1ifandonly ifthepair( A, B )hasdistinctcoecientsfor(1 Š T ),andweobtain = 0 2+ 2 0+ 1 2+ 2 1+ 0 3+ 3 0+ 1 3+ 3 1. Thecocycles 0= 2 1+ 2 3, 1= 1 0+ 1 2,and asconstructedinExample4.1.4(1)arelinearlyindependent(evalu ateonthecyclesde“nedinRemark4.1.9), and = + 0+ 1. Abelianextensionsde“nesurjectivehomomorphisms E ( X,A, )= A X X de“nedbytheprojectionontothesecondfact or.Itwasprovedin[11]thattwoabelian extensions E ( X,A, )and E ( X,A,)areisomorphicifandonlyif iscohomologous to Proposition4.1.5 Thecocycles obtainedfromtheabelianextensions Zqm +1[ T,TŠ 1] / ( T Š 1+ q )= E ( Zqm[ T,TŠ 1] / ( T Š 1+ q ) Zq, ) Zq[ T,TŠ 1] / (1 Š T )m +1= E ( Zq[ T,TŠ 1] / (1 Š T )m, Zq,) respectively,arenotcoboundaries. Proof. Directcomputationsshowthatthechains c =(0 1)+( q,qm Š 1+ q Š 1) ZQ 2( X ; Zq)and c=(0 1)+(1 Š T, (1 Š T )m Š 1+(1 Š T ) Š 1) ZQ 2( X ; Zq) arecyclesfor X = Zqm[ T,TŠ 1] / ( T Š 1+ q )and X = Zq[ T,TŠ 1] / (1 Š T )m,respectively. Then,itiscomputedthat ( c )=1and ( c)=1,andhence and arenot coboundaries,andtheresultfollows. 40

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Anextensiontheoryofquandlesfortwistedcohomologycocycleswasdeveloped in[6],anditprovidedmoregeneralexte nsiontheories.Inthetwistedcase,the coecientgroupistakentobea-module, thushasanAlexanderquandlestructure andtheextension AE ( X,A, )=( A X, )isde“nedby( a1,x1) ( a2,x2)=( a1 a2+ ( x1,x2) ,x1 x2)for Z2 TQ( X ; A ),andiscalledan Alexanderextension of X by( A, ).Forexample, RpeisanAlexanderextensionof Rpe Š 1by Rpsuchthat Rpe= AE ( Rpe Š 1,Rp, ),forsome Z2 TQ( Rpe Š 1; Rp). Remark4.1.6 Thequandlestructureofadihedralquandle Rnisde“nedusingthe ringstructureof Zn.Theproductquandle Rm Rnisde“nedbycomponent-wiseoperation,sothatitisde“nedfromtheringstructureof Zm Znaswell.Consequently, twoquandles Rm Rnand Rmnareisomorphicif Zm Znand Zmnareisomorphic asrings.Hence,if n = pe11 pekkistheprimedecomposition,then Rnisisomorphic to Rpe 1 1 Rpe k k.For p =2,theresultofthissectionshowsthat Rpeisdescribed succesivelyasanextensionof Rpe Š 1. Ohtsuki[37]de“nedanextensiontheoryandanewcohomologytheoryforquandles,togetherwithalistofproblemsinthesubject. Thefollowinglemmafollowsfromde“nitions. Lemma4.1.7 [5] Let X,Y bequandlesand A beanabeliangroup.If E isanabelian extensionof X for Z2 Q( X ; A ) : E = E ( X,A, ) ,then E Y isanabelianextension of X Y for p# Z2 Q( X Y ; A ) : E Y = E ( X Y,A,p# ) ,where p : X Y X istheprojectiontotherstfactor. Corollary4.1.8 Foranypositiveinteger n E = R4 nisanabelianextension E = E ( R2 n, Z2, ) of X = R2 nforsomecocycle Z2 Q( R2 n; Z2) Proof. Let2 n =2mk foranoddinteger k .Then R2 n = R2m RkbyRemark4.1.6,and byLemma4.1.7, R4 n = R2m +1 Rkisanabelianextensionof R2 nif R2m +1isanabelian 41

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extensionof R2m.ThisfollowsfromTheorem4.1.2since R2m = Z2m[ T,TŠ 1] / ( T +1). Remark4.1.9 ByLemma4.1.1andCorollary4.1.8,thereisacocycle Z2 Q( R4 n; Z2) suchthat R8 nisisomorphicto E ( R4 n, Z2, ). Let 0 1,1 0 Z2 Q( R4 n; Z2)becocyclesde“nedby 0 1= p#( 0 1+ 0 3) and 1 0= p#( 1 0+ 1 2) respectively,where p : R4 n R4isanaturalmap p ( x mod(4 n ))= x mod(4).Here, itisknown[8]that 0 1+ 0 3, and 1 0+ 1 2arecocyclesin Z2 Q( R4; Z2).Itisdirectlycomputedthat c0 1=(0 1)+(2 1) ,c1 0=(1 0)+(4 n Š 1 0) ,c 0 1=(0 1)+(2 2 n +1) ZQ 2( R4 n; Z2) arecycles.Then,wehave 0 1( c0 1)=1 ,0 1( c1 0)=0 ,0 1( c 0 1)=1 1 0( c0 1)=0 ,1 0( c1 0)=1 ,1 0( c 0 1)=0 ( c0 1)=0 ( c1 0)=0 ( c 0 1)=1 where iscomputedinExample4.1.4.Hence,weseethatthecocycles 0 1,1 0,and arelinearlyindependent. Lemma4.1.10 [5] Let X beaquandle, i Z2 Q( X ; A ) i =0 1 ,where A isan abeliangroup,and E1= E ( X,1) beanextension,andlet p : E1 X bethenatural homomorphism(theprojectionontothesecondfactor).Then, E0= E ( E1,A,p#0) isisomorphicto E2= E ( X, ( A,0) ( A,1)) 42

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Proof. Assets, E0= A E1= A ( A X )and E2= A A X .Let f : E0 E2be theidentityŽmapbetweenthesesets.Weshowthat f isanisomorphism.Itonly needstobecheckedthat f isahomomorphism.Onecomputes f (( a1, ( b1,x1)) ( a2, ( b2,x2))) = f ( a1p0(( b1,x1) ( b2,x2)) ( b1,x1) ( b2,x2)) = f ( a10( p ( b1,x1) ,p ( b2,x2)) ( b11( x1,x2) ,x1 x2)) = f ( a10( x1,x2) (( b11( x1,x2) ,x1 x2)) =( a10( x1,x2) ,b11( x1,x2) ,x1 x2) = f (( a1, ( b1,x1))) f (( a2, ( b2,x2))) asdesired. Lemma4.1.11 Let p : R4 n T2bethequotienthomomorphismdenedby p ( x )= x (mod2) ,and Z2 Q( T2; Z2) .Then, (a) E ( R4 n, Z2,p ) hasasubquandleisomorphicto T4,and (b) E ( R4 n, Z2,p ) isnotisomorphicto R8 n. Proof. ByLemmas4.1.8and4.1.10, E = E ( R4 n, Z2,p )isisomorphicto E0= E ( R2 n, ( Z2, ) ( Z2, )),since R4 n= E ( R2 n, Z2, )forsome Z2 Q( R2 n; Z2).Then, thesubset { ( x,y, 0) Z2 Z2 R2 n= E0} formsasubquandleisomorphicto T4.Thisproves(a). Toprove(b),itissucienttoprovethat R8 ndoesnotcontainasubquandle isomorphicto T4.If a b = a R8 n,then2 b Š a a (mod4 n ),hence a b (mod2 n ). However,thereareonlytwosuchintegersmod4 n .Hence,thelargestsubquandle isomorphictoatrivialquandlehastwoelements. 43

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4.2Evaluationsofcocycleinvariantsusingextensioncocycles Asexamples,weevaluatethepreviously de“nedinvariantsforWhiteheadlinkand Borromeanrings,usingextensioncocyclesconstructedinSection4.1.Weusethe coecientgroup A = Zq= { tn| n =0 1 ,...,q Š 1 } forapositiveinteger q .2(d, e) (a, b)(b, c)(e, d)(e, c)(a, e)_ ab a*b=c b*(a*b)=d KK b*(b*(a*b))=e1Figure15.AcoloredWhiteheadlink Example4.2.1 InFigure15,aWhiteheadlink L = K1 K2isdepicted.Let Z2 Q( R8; Z2)bethecocyclede“nedbyCorollary4.1.8.Weevaluatethecomponentwisecocycleinvariant ( L )=(1( L ) 2( L )),whichwasgiveninDe“nition3.3.1. Denotethemultiplicativegeneratorofthecoecientgroup Z2by t ,sothat Z2= { 1 ,t } andtheinvarianttakestheform ( L )=( A1+ B1t,A2+ B2t ),where Ai,Bi, i =1 2, arenon-negativeintegers. Thecolorsassignedtoarcsarerepresentedbytheletters a through e .Fromthe “gure,itisseenthatallthecolorsaredeterminedbythecolors a and b assigned tothetoptwoarcs.Weobservefromthecalculationsthatforanychoiceoftwo elements a and b of R8,thereisauniquecoloringof L by R8thatrestrictstothe chosenelements a and b .Therefore,thereare82=64coloringsof L by R8. 44

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Weshowthatthestate-sumterm T1B ( C )istrivialifandonlyif a and b havethesameparity(bothevenorbothodd). Supposethat a and b arebotheven,sothat a =2 b =2 .Then,onecomputes that c =4 Š 2 d =6 Š 4 ,andweobtain e =2 = b .Similarcomputationsshow that e = b ,if a and b arebothodd.Fromthe“gure,thestate-sumtermfor T1is ( a,b ) Š ( a,e ),whichisequalto ( a,b ) Š ( a,b )=0,inthiscase.Supposenowthat a and b haveoppositeparities.Bysetting a =2 +1and b =2 (andviceversa), wecomputethat e = b +4,sothatweobtainthestate-sumterm ( a,b ) Š ( a,e )= ( a,b ) Š ( a,b +4).Weclaimthatthisis t UsingtheformulaattheendoftheproofofTheorem4.1.2,wehave ( a,b )= [ s ( a ) s ( b ) Š s ( a b )] / 8.Here, s ( a ) s ( b )is2 b Š a computedmodulo16,and s ( a b )is 2 b Š a computedmodulo8,thenregardedasanelementmodulo16.Since a ( b +4)= 2( b +4) Š a =(2 b Š a )+8modulo16,wehave ( a,b ) Š ( a,b +4)=[ s ( a ) s ( b ) Š s ( a b )] / 8 Š [ s ( a ) s ( b )+8 Š s ( a b )] / 8=1(mod2), writtenadditively.Thisprovestheaboveclaim.Thereare32coloringswiththesame parity,and32withdistinctparities.Hence,weobtain ( L )=(32+32 t, 32+32 t ). Thefollowingexamplesdealwiththegeneralizedde“nitionsofcocycleinvariants asthoseweredescribedinDe“nitions3.3.1and3.3.3. Example4.2.2 Let X = Wm= Zq[ T,TŠ 1] / (1 Š T )mor X = Um= Zqm[ T,TŠ 1] / ( T Š 1+ q ),and L theWhiteheadlink.Thenthegeneralizedcocycleinvariantis ( L )= { (1 1) ,..., (1 1) q2 mcopies} for m =1 2, { ( tn,tŠ n) ,..., ( tn,tŠ n) qm +2copies}n { 0 1 ,...,q Š 1 }for m 3. 45

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25 3 421 KK3b w2w w w1* w w2 1b5 4 6 1 6Figure16.Whiteheadlink Consequently, ( L )= ( q2 m,q2 m)for m =1 2, ( qm +2( tq Š 1+ + t +1) ,qm +2( tq Š 1+ + t +1))for m 3. Proof. Let X = Zq[ T,TŠ 1] / (1 Š T )m.Thecasefor X = Zqm[ T,TŠ 1] / ( T Š 1+ q ) issimilar.Pickbasepoints b1and b2onthecomponents K1and K2,respectively,of theWhiteheadlink L = K1 K2asdepictedinFigure16,andtraceeachcomponent inthegivenorientationofthelink.Thecolorsassignedtothearcsareelementsof X andappearinthisorder w1,w2for K1,and w3,...,w6for K2asshowninthe “gure.Thecrossingatthetailofthearccoloredby wiisde“nedtobe i.First, wedeterminethesetofcolorings:For m 3andfortwoelements w1, w3 WmassignedtothetoptwoarcsoftheWhiteheadlink L ,thereisacoloringof L by X whichrestrictstothegiven w1, w3ifandonlyif w3Š w1 0(mod(1 Š T )m Š 3) 46

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For m =1 2,thereissuchacoloringforany w1,w3 X .Thiscanbecomputedas follows. Representtheelementsof X = Zq[ T,TŠ 1] / (1 Š T )mby a = am Š 1(1 Š T )m Š 1+ + a1(1 Š T )+ a0,where aj Zq.Notethat(1 Š (1 Š T ))(1+(1 Š T )+ +(1 Š T )m Š 1)=1 in X ,so TŠ 1=1+(1 Š T )+ +(1 Š T )m Š 1.Notealsothat a b = TŠ 1a +(1 Š TŠ 1) b Wehavethefollowingcalculationsforeacharc: w2= w1 w3= w1+(1 Š T )( w3Š w1) w4= w3 w2= w3+(1 Š T )( w2Š w3) =( w3Š w1)(1 Š T )2Š ( w3Š w1)(1 Š T )+ w3, w6= w3 w4= TŠ 1w3+(1 Š TŠ 1) w4=( w3Š w1)(1 Š T )2+ w3, w5= w4 w6= w4+(1 Š T )( w6Š w4) =2( w3Š w1)(1 Š T )2Š ( w3Š w1)(1 Š T )+ w3. Theserelationsareobtainedusingthetopfourcrossings( 2,4,3,and 5).The bottomtwocrossings( 6and 1)ofthelinkgiverisetorelations.The“rstrelation is w6 w2= w5forthesecondbottomcrossing,giving( w1Š w3)(1 Š T )3 0 (mod(1 Š T )m).Thesecondrelationthatcorrespondstothebottomcrossingis w1 w6= w2giving( w3Š w1)(1 Š T )3 0(mod(1 Š T )m),asclaimedabove. Nowwedeterminethecontributiontotheinvariantforeachcoloring.Recallthat ( w1,w3)=[ s ( w1) s ( w3) Š s ( w1 w3)] / (1 Š T )m.Since( w3Š w1)(1 Š T )3 0 (mod(1 Š T )m),weseethatthecontributionis ( w1,w3) Š ( w1,w6) =[ s ( w1) s ( w3) Š s ( w1 w3)] / (1 Š T )mŠ [ s ( w1) s ( w3)+( w3Š w1)(1 Š T )3Š s ( w1 w3)] / (1 Š T )m47

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= Š ( w3Š w1)(1 Š T )3/ (1 Š T )m(mod q ) forthe“rstcomponent,andforthesecondcomponent,computationsshowthat ( w3,w2) Š ( w6,w2)+ ( w6,w4)+ ( w4,w6)=( w3Š w1)(1 Š T )3/ (1 Š T )m(mod q ) For m =1 2thecontributionsforthe“rstandthesecondcomponentareboth0, andwehave qmchoicesforboth w1and w3,therefore ( L )=((1 1) ,..., (1 1) q2 mcopies)and ( L )=( q2 m,q2 m). For m 3,if w1and w3color L ,then( w3Š w1)(1 Š T )3is0asanelement of X ,sothat w3Š w1isuniquelywrittenas w3Š w1= k (1 Š T )m Š 3,where k = k0+ k1(1 Š T )+ k2(1 Š T )2,and k0,k1,k2{ 0 1 ,...,q Š 1 } .Then ( w3Š w1)(1 Š T )3= k (1 Š T )m=( k0+ k1(1 Š T )+ k2(1 Š T )2)(1 Š T )m= k0(1 Š T )m Wm +1. Thus,thecontributiontotheinvariantforthe“rstandsecondcomponentsare tŠ k0and tk0,respectively. To“ndthenumberofcoloringsconrtibutingto tŠ k0and tk0,“x k0.Wehave qmchoicesfor w1and q2choicesfor k .Then, w3isuniquelydeterminedby w3= w1+ k (1 Š T )m Š 3.Intotal,thecontributionis qmq2= qm +2foreach tŠ k0and tk0. Setting n = Š k0weobtaintheresult. 48

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K2K3 1y K1y3 4 6y 2y 5y yFigure17.Borromeanrings Example4.2.3 Let X = Wm= Zq[ T,TŠ 1] / (1 Š T )mor X = Um= Zqm[ T,TŠ 1] / ( T Š 1+ q ),and L theBorromeanrings.Then,thegeneralizedcocycleinvariantis ( L )= { (1 1 1) ,..., (1 1 1) q3 mcopies} for m =1, { ( tŠ k0,tŠ 0,tk0+ 0) ,..., ( tŠ k0,tŠ 0,tk0+ 0) qm +2copies}k0,0{ 0 1 ,...,q Š 1 }for m 2. Consequently, ( L )= ( q3 m,q3 m,q3 m)for m =1, ( qm +2( tq Š 1+ +1) ,qm +2( tq Š 1+ +1) ,qm +2( tq Š 1+ +1)) for m 2. Proof. Let X = Wmandlet L betheBorromeanringsasdepictedinFigure17. Thecasefor X = Umissimilar.Calculationsaresimilartotheprecedingexample andwegiveasketch.First,wedeterminethesetofcolorings:Forthreeelements y1,y2,y3 X assignedtoeachouterarcinthediagramof L ,thereisacoloringof L by X whichrestrictstothegiven y1,y2,y3ifandonlyif ( y2Š y3)(1 Š T )2 0 ( y1Š y2)(1 Š T )2 0and( y3Š y1)(1 Š T )2 0 49

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Theouterthreecrossingsareusedtodescribe y4,y5,y6intermsof y1,y2,y3,andthe innerthreecrossingsgivetheaboverelations. Contributionstotheinvariantarecomput edasfollows.Thecontributionforthe “rstcomponentof L coloredby y1is ( y1,y2) Š ( y1,y2 y3)= Š ( y3Š y2)(1 Š T )2/ (1 Š T )m(mod q ).For m =1,thecontributionistrivial,andthetotalnumber ofcoloringsis q3 m.For m 2,( y3Š y2)(1 Š T )2isdivisibleby(1 Š T )m,so y3Š y2isuniquelywrittenas y3Š y2= k (1 Š T )m Š 2,where k = k0+ k1(1 Š T )and k0,k1 { 0 1 ,...,q Š 1 } .Therefore, ( y3Š y2)(1 Š T )2= k (1 Š T )m=( k0+ k1q )(1 Š T )m= k0(1 Š T )m, andthe“rstcomponentcontributes tŠ k0totheinvariant.Forthesecondcomponentof L coloredby y2,similarcalculationsasabovegivethecontribution ( y2,y3) Š ( y2,y3 y1)= Š ( y1Š y3)(1 Š T )2,whichisdivisibleby(1 Š T )mso y1Š y3=( 0+ 1(1 Š T ))(1 Š T )m Š 2andtherefore Š ( y1Š y3)(1 Š T )2= Š 0(1 Š T )m.Theweobtain y2Š y1= Š [( k0+ 0)+( k1+ 1)(1 Š T )](1 Š T )m Š 2,sothatthethirdcomponentcontributes tk0+ 0.Finally,thecontributiontotheinvariantisthevector( tŠ k0,tŠ 0,tk0+ 0),where theentriescorrespondtothecomponents K1,K2,K3,respectively.Theresultfollows. 50

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CHAPTER5 EXTENDINGCOLORINGSOFKNOTS InChapter2,wedemonstratedhowaknotdiagramcanbecoloredbyaquandle. Furthermore,inChapter4weintroducedthenotionofextensionsofquandles.Since extensionsofquandlesarealsoquandles ,weareledtoask:whencanacoloringof aknotbyaquandlebeextendedtoacoloringbyanextensionofthequandle?We investigatethisprobleminthefollowingchapter. 5.1Extensionsofcolorings Denition5.1.1 Let K beaclassicalknotorlink.Let C beacoloringof K by X Let E beaquandlewithasurjectivehomomorphism p : E X .Ifthereisacoloring Cof K by E ,suchthatforeveryarc a of K itholdsthat p ( C( a ))= C ( a ),then Cis calledan extension of C Example5.1.2 Let K bethe(4 2)-toruslinkshowninFigure14.Let X = R4and let E ( R4, Z2, )= R8,where isconstructedinExample4.1.4(1).Notethatany pairofelementsof X assignedto aiand bk,uniquelydeterminethecolorsassigned toallthearcsinthediagram.Let C beacoloringby R4determinedby ai=0and bk=2.Thus, aj=0and b=2.Observethat ai=0and bk=2alsouniquely determineacoloringby R8,with aj=4and b=6.Callthiscoloring C.Then, Cisanextensionof C Nowlet C bethecoloringof K by R4determinedby ai=0and bk=1.Let ai=0 and bk=1in R8.Then,thetoptwocrossingsrequirethat aj=2and b=3in R8. Thisassignmentthoughdoesnotsatisfytherequirementofacoloringforthebottom 51

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twocrossings,andtherefore C doesnotextendtoacoloringby R8forthischoice.By checkingallthepossibilities( ai,bk)= { (0 1) (0 5) (4 1) (4 5) } weconcludethat C doesnotextend. Bycheckingallthecases,we“ndthatacoloring C by R4extendstoacoloring Cby R8ifandonlyifthepair( ai,bk)isfromtheset { (0 0) (0 2) (1 1) (1 3) (2 0) (2 2) (3 1) (3 3) } 5.2Cocycleknotinvariantsasobstructionstoextendingcolorings Let K beaknotanddenoteby( K )thestate-suminvariantof K ,aswasgivenin De“nition3.2.1,withrespecttoaquandle X ,anabeliangroup A ,andacocycle Z2 Q( X ; A ).Let E = E ( X,A, )betheabelianextensionof X by .Wecharacterize whenthestate-suminvariantde“nedfromthiscocycleisnon-trivial,ifthecocycles usedarethosede“nedfromabelianextensions. Forcharacterizationsonthetriviality ofcolorings,see[24]. Theorem5.2.1 [5] Let C0( K,X ) betheconstantterm(apositiveinteger)of ( K ) and C ( K,X ) bethenumberofallcoloringsof K by X .Then,thenumberofcolorings of K by X thatextendtocoloringsof K by E ( X,A, ) isequalto C0( K,X ) ,andthe numberofcoloringsthatdonotextendis C ( K,X ) Š C0( K,X ) Proof. Let C beacoloringwhosecontributionto( K )is1.Fixthiscoloringinwhat follows.Pickabasepoint b0onaknotdiagramof K .Let x X bethecoloronthe arc 0containing b0.Let i, i =1 ,...,n ,bethesetofarcsthatappearinthisorder whenthediagramof K istracedinthegivenorientationof K ,startingfrom b0.Pick anelement a A andgiveacolor( a,x )on 0,sothatwede“neacoloring Cby E on 0by C( 0)=( a,x ) E .Wetrytoextendittotheentirediagrambytraveling thediagramfrom b0alongthearcs i, i =1 ,...,n ,inthisorder,byinduction. 52

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Assume C( i)isde“nedfor0 i
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fortheknots 31, 41, 72, 73, 81, 84, 811, 813, 91, 912, 913, 914, 921, 923, 935, 937, andthevalue16+48 t for818, 940.Hence,fortheknotsintheformerlist,thenumber ofcoloringsby X whichextendtothoseby E is4(trivialcolorings,byasinglecolor), andthosethatdonotextendis12(allnon-trivialcoloringsdonotextend).Forthe knotsinthelatterlist,thereare16coloringsthatextend,and48coloringsthatdo not. Corollary5.2.3 For A = Z2andsomecocycle Z2 Q( X ; A ) ( K )= a + bt ,where t isthevariable,i.e.thegeneratorof Z2,isdeterminedbythenumberofcolorings with X and E : a isthenumberofcoloringsof X thatextendtocoloringsby E ,and b isthenumberofthosethatdonot. Let L = K1 Krbealinkdiagram.RecallfromDe“nition3.3.1thegeneralizationofthestate-suminvarianttolinkscomponent-wise.Weobserveherethat Theorem5.2.1inthissectionappliestoc omponent-wisecocycleinvariants. Theorem5.2.4 Let ( L )=(i( L ))r i =1bethecomponent-wisecocycleinvariantofa link L = K1 Krwithaquandle X andacocycle Z2 Q( X ; A ) foranabelian group A .Then, i( L ) isnotapositiveintegerforsome i ifandonlyifthereisa coloringof L by X thatdoesnotextendtoacoloringof L by E ( X,A, ) Example5.2.5 Let L beacoloredWhiteheadlink L = K1 K2,asdepictedin Figure15.WehaveseeninExample4.2.1thatthecomponent-wisecocycleinvariant is ( L )=(32+32 t, 32+32 t ),wherethecocycle de“nestheextension E = R16= E ( R8, Z2, ). Theorem5.2.4impliesthattherearecoloringsby R8thatdonotextendtocoloringsby R16.Infact,fromtheproofofTheorem5.2.1,weseethat32coloringshaving 54

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thesameparityfor a and b extendto R16,andthose32coloringswiththeopposite paritiesdonot.Thisfactcanbecomputed directly,andgivesanalternatemethod ofcomputingtheaboveinvariantusingCorollary5.2.3. Example5.2.6 Let L bethe(4 2)-toruslink(seeFigure14)coloredbythequandle X = R4.Theextension E = R8= E ( R4, Z2, )isde“nedbythecocycle (see Example4.1.4(1)),where = 0 2+ 0 3+ 1 0+ 1 3+ 2 0+ 2 3+ 3 0+ 3 1. Thecomponent-wiseinvariantiscomputedtobe ( L )=(8+8 t, 8+8 t ).Then, byTheorem5.2.4wecanassertthatthereare8coloringsby R4thatextendto coloringsby R8,and8thatdonotextend.Thecoloringsthatdoextendarelistedin Example5.1.2. Fromtheexamples4.2.2and4.2.3,weseethatthecocycleinvariantisnontrivialwhenthegivenlinkiscoloredby X = Zq[ T,TŠ 1] / (1 Š T )m,butnotby E = Zq[ T,TŠ 1] / (1 Š T )m +1,andthedescrepancyinextendingthecoloringcontributes totheinvariant.Thisisthecaseingeneral,asprovedin[5]fortheknotcase. Werephrasethetheoreminoursituationandincludeasimilarproofforreaders convenience. Theorem5.2.7 [7]Let ( L )= T1B ( C ) ,..., TrB ( C ) C ColX( L )be thegeneralizedcocycleinvariantofalink L = K1 Krwithaquandle X andacocycle Z2 Q( X ; A ),foranabeliangroup A .Then, T1B ( C ) ,..., TrB ( C ) isavectorwitheveryentry1foracoloring C ifandonlyifthecoloring C extendsto acoloringof L by E ( X,A, ). Proof. Let C beacoloringwhosecontributionto ( L )is(1 ,..., 1).Fixthis coloringinwhatfollows.Pickabasepoint b0onacomponent Kiof L .Let x X bethecoloronthearc 0containing b0.Let i, i =1 ,...,n ,bethesetofarcsthat appearinthisorderwhenthediagram K istracedinthegivenorientationof Ki, 55

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startingfrom b0.Pickanelement a A andgiveacolor( a,x )on 0,sothatwe de“neacoloring Cby E on 0by C( 0)=( a,x ) E .Wetrytoextendittothe entirediagrambytravelingthediagramfrom b0alongthearcs i, i =1 ,...,n ,in thisorder,byinduction. Assume C( i)isde“nedfor0 i
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CHAPTER6 RELATIONSTOALEXANDERMATRICES ThediscoveryoftheAlexandermatrixandtheAlexanderpolynomialwasoneof theearlyachievementsinknottheory.Inthischapter,wepointoutrelationsofthe cocycleinvariantstoAlexandermatrices.Weexaminecloselythetwoexamplesgiven inSection3.3fromthisnewpointofview. 6.1CocycleinvariantsandAlexandermatrices Foralinkdiagram DLlet BDL= n i =1Bibean( n n )-matrix,where Biisthe( n n )matrixcorrespondingtoeachcrossingpoint i(seeFigure15)suchthatthe( ki,i ) entryis Ti,the( i,i )entryis1 Š Tiandotherwiseis0.Here, idenotesthesign ofthecrossingpoint i.Set ADL= BDLŠ En,where Endenotesthe n -dimensional identitymatrix.Itfollowsfromthede“nitions[24]that ADLisanAlexandermatrix. Recallthatacoloringisafunction C : R X ,where R isthesetofover-arcsinthe diagramand X isa“xedAlexanderquandle /J foranideal J .Acoloringwhich assigns witoanarc ai( C ( ai)= wi)isrepresentedbythevector w =( w1,...,wn) satisfying wA( X ) DL= 0.Thesedescriptionsaregivenin[24]toproveTheorem2.4.4. Proposition6.1.1 Let L = K1 Krbealinkand X =q/J beanAlexander quandle.Suppose E =q/Jisanabelianextensionof X ,where q,qarepositive integers.Let A( X ) DL(respectively A( E ) DL)bethematrix ADLregardedasamatrixover X (respectivelyover E ).Then,acoloring w of L by X contributesanon-trivialvalueto theinvariant ( L ) ifandonlyif wA( X ) DL= 0 and s ( w ) A( E ) DL= x = 0 ,where s : X E isaset-theoreticsection. 57

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wli wiwkiiFigure18.Labelingacrossing Proof. Let :(q/J )n (q/J )nbethemapwhichtakesarowvector w to wADL. ByInouesdescriptiongivenabove,thesetofallquandlecoloringsisequaltoker A( X ) DL. If wA( X ) DL= 0and s ( w ) A( E ) DL= x =0,thenbyTheorem5.2.7weobtainthat ( L )is non-trivial. Next,wecomputethenon-trivialcontribut ionsusingAlexandermatrices,forthe extensionsdiscussedinSection4.1.Let X = Wm=q/ (1 Š T )mor X = Um= qm/ ( T Š 1+ q ),andlet E = Wm +1or E = Um +1betheirabelianextensions, respectively.Forthispurpose,we“xthefollowingconventioninnumberingcrossings andarcsofagivendiagram. Let L = K1 Krbealinkwith n crossings.Pickabasepoint bion Ki,for i =1 ,...,r .Let a1,...,ai1bethearcsof K1suchthat a1contains b1andthatthey appearinthisorderwhenonetraces K1inthegivenorientationof K1startingfrom b1.Then,let ai1+1bethearcof K2containing b2and ai1+2,...,ai2bethearcsof K2similarlyde“nedfromthegivenorientation.Repeatthisprocessfortheremaining componentstoobtainthearcs a1,...,ai1,ai1+1,...,ai2,ai2+1,...,air Š 1+1,...,air= an.Let C : R X beacoloringof L by X .Let wi= C ( ai)and ibethe crossingsuchthattheoutcomingunder-arcis aifor i =1 ,...,n (seeFigure18).This conventionisusedinFigure15. Let s : X E bethesectionde“nedinSection4.1by s m Š 1j =0Aj(1 Š T )jmod(1 Š T )m =m Š 1j =0Aj(1 Š T )jmod(1 Š T )m +1for Wm, 58

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and s m Š 1j =0Xjqj =0 qm+m Š 1j =0Xjqjfor Um. Forthefollowingproposition,let ( L )bethegeneralizedcocycleinvariantde“ned withthecocycle Z2 Q( X ; Zq)correspondingtotheextension p : E X speci“ed byDe“nition3.3.3. Proposition6.1.2 Let ADLbetheAlexandermatrixobtainedfromadiagram DLwiththeabovechoiceoforderof wiand i. Agivencoloringrepresentedbyavector w contributesanon-trivialvectortothe invariant ( L ) ifandonlyif wA( X ) DL= 0 and s ( w ) A( E ) DL= z = 0 .Thiscontributionis ( ti 1 j =1 ( j) zj/ (1 Š T )m,...,ti r j = i r Š 1 +1 ( j) zj/ (1 Š T )m)for X = Wm, and ( ti 1 j =1 ( j) zj/qm,...,ti r j = i r Š 1 +1 ( j) zj/qm)for X = Um, respectively,where ( )=1 forapositivecrossing and ( )= T foranegative crossing Proof. Weconsiderthecase X = Wm,astheothercaseissimilar.Let :(q/ (1 Š T )m)n (q/ (1 Š T )m)nbethemapwhichtakesarowvector w to wA( X ) DL.Assume that wA( X ) DL= 0and s ( w ) A( E ) DL= z = 0.Thecontributiontotheinvariantatapositive crossing iisgivenby ( wki,wi)=[ s ( wki) s ( wi) Š s ( wki wi)] / (1 Š T )m=[ s ( wki) s ( wi) Š s ( wi)] / (1 Š T )m, 59

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where wiisthecolorontheover-arcatthecrossing i,and wkiisthecoloron theincomingunder-arcat iif iispositive(seeFigure18).Since w isinthe kernel, wki wiŠ wi= Twki+(1 Š T ) wiŠ wi=0mod(1 Š T )mandwehave [ s ( wki) s ( wi) Š s ( wi)] / (1 Š T )m= zi/ (1 Š T )m. Suppose iisnegative.Then,thecontributionis Š ( wi,wi)= Š [ s ( wi) s ( wi) Š s ( wi wi)] / (1 Š T )m= Š [ s ( wi) s ( wi) Š s ( wki)] / (1 Š T )m= Š [ Twi+(1 Š T ) wiŠ wki] / (1 Š T )m. Ontheotherhand, zi= TŠ 1wki+(1 Š TŠ 1) wiŠ wi= Š TŠ 1[ Twi+(1 Š T ) wiŠ wki] sothatthecontributionis Tzi,inthiscase.Hence,thetotalcontributionofthe invariantforthecomponent Kris ti r j = i r Š 1 +1 ( j) zj/ (1 Š T )m, where { z1,...,zir} Kr. Example6.1.3 WeconsidertheWhiteheadlink L = K1 K2depictedinFigure15. Let X = Wmand E = Wm +1.Usetheletters wi( i =1 ,..., 6)asshowninthe“gure ascolorsassignedtothearcs,aswellasge neratorsfortheAlexandermatrix.Then, theAlexandermatrix ADL= BDLŠ Enwithrespecttothecolumnscorresponding to( 1,...,6)androwscorrespondingto( w1,...,w6)isgivenby 60

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ADL= Š 1 T 0000 TŠ 1Š 101 Š T 01 Š TŠ 101 Š T Š 1 T 00 001 Š T Š 1 T 0 0000 Š 1 TŠ 11 Š TŠ 10 T 01 Š T Š 1 Aftersomerowandcolumnpermutationsweobtain A0= Š 11 Š T 001 Š TŠ 1TŠ 10 Š 11 Š TT 00 00 T 1 Š T Š 11 Š TŠ 1000 Š 1 TŠ 10 T 0000 Š 1 1 Š TT Š 1000 withrespecttothecolumnscorrespondingto( 2,4,3,5,6,1)androwscorrespondingto( w2,w4,w6,w5,w1,w3).Thispermutationisperformedsothatwecan diagonalizethe“rstfourrowsandcolumnsbycolumnreductionstoobtain A1= 100000 010000 001000 000100 Š T Š T + T2(1 Š T )21 Š 3 T +2 T2Š TŠ 1(1 Š T )3TŠ 1(1 Š T )3Š 1+ T Š 1+ T Š T2Š 1 Š (1 Š T )2Š 2+3 T Š 2 T2TŠ 1(1 Š T )3Š TŠ 1(1 Š T )3 61

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Thesolutionset ( w2,w4,w6,w5,w1,w3) A( X ) 1=(0 0 0 0 0 0) iswrittenby w2= Tw1+(1 Š T ) w3w4= T (1 Š T ) w1+( T +(1 Š T )2) w3w6= Š (1 Š T )2w1+(1+(1 Š T )2) w3w5=( T (1 Š T ) Š (1 Š T )2) w1+( T +2(1 Š T )2) w30=( w3Š w1) TŠ 1(1 Š T )3where A( X ) 1denotesthematrix A1regardedasamatrixover X .Thesetofcolorings isrepresentedbyvectorsinthekernelof A( X ) 1.Speci“cally,thekernelisthesetof vectors w with w1and w3satisfying(1 Š T )3( w3Š w1)=0in X and w2,w4,w6,w5determinedaccordinglyasabove.Thismat chesthecomputationsinExample4.2.2. Thecontributiontotheinvariantisobtainedbycomputing z = s ( w ) A( E ) DL=( Š TŠ 1(1 Š T )3( w3Š w1) 0 0 0 0 ,TŠ 1(1 Š T )3( w3Š w1)) ByProposition6.1.2,thenon-trivialcontributionto ( L )is ( t2 j =1 ( j) zj/ (1 Š T )3,t6 j =3 ( j) zj/ (1 Š T )3)=( tŠ s,ts) forsome s ,for0 s q Š 1,dependingonthevalueof w3Š w1.Thisresultmatches theoneinExample4.2.2. 62

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Example6.1.4 Let L = K1 K2 K3betheBorromeanringsasdepictedinFigure17.TheAlexandermatrix ADL,wheretherowscorrespondtothecrossings 1,...,6,andthecolumnscorrespondto y1,...,y6fromlefttoright,respectively,is givenbythematrix ADL= Š 1001 Š TT 0 0 Š 1001 Š TT 00 Š 1 T 01 Š T 01 Š TŠ 1TŠ 1Š 100 TŠ 101 Š TŠ 10 Š 10 1 Š TŠ 1TŠ 1000 Š 1 Thevector y has3independententries y1,y2,y3,andtheotherthreeentries y4,y5,y6arelinearcombinationsofthese.Thesolutionvectorisgivenby y1,y2,y3,y4=(1 Š T ) y1+ Ty3,y5= Ty1+(1 Š T ) y2,y6= Ty2+(1 Š T ) y3 Thesolutionset y canalsobeobtainedbycolumnreductionsaswasdoneinthe previousexample.Byrearrangingrow sandcolumns,weobtainanewmatrix A1in suchawaythattherowsof A1correspondto( y4,y5,y6,y1,y2,y3)andthecolumnsof A1correspondto( 4,5,6,1,2,3)sothat A1= Š 10001 Š TŠ 1TŠ 10 Š 10 TŠ 101 Š TŠ 100 Š 11 Š TŠ 1TŠ 10 1 Š TT 0 Š 100 01 Š TT 0 Š 10 T 01 Š T 00 Š 1 63

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Bythisrearrangement,wecandiagonalizethe“rstthreerowsandcolumnsbycolumn reductionstoobtain A2= 100000 010000 001000 T Š 1 Š T 00 Š TŠ 1(1 Š T )2TŠ 1(1 Š T )20 T Š 1 Š TTŠ 1(1 Š T )20 Š TŠ 1(1 Š T )2Š T 0 T Š 1 Š TŠ 1(1 Š T )2TŠ 1(1 Š T )20 Hence,thesolutionsetiswrittenby ( y4,y5,y6,y1,y2,y3) A( X ) 2=(0 0 0 0 0 0) andweobtain y4=(1 Š T ) y1+ Ty3, y5= Ty1+(1 Š T ) y2, y6= Ty2+(1 Š T ) y3, y1, y2, y3 freevariables. Then, s ( y ) ADL= x ,wherethevector x isgivenby x = TŠ 1(1 Š T )2( y2Š y3) ,TŠ 1(1 Š T )2( y3Š y1) ,TŠ 1(1 Š T )2( y1Š y2) 0 0 0 The“rststatementofTheorem6.1.1(th ecoloringcondition)impliesthat L iscolored by X =q/J ifandonlyif x = 0in X ,i.e.,(1 Š T )2( y2Š y3)=0 (1 Š T )2( y3Š y1)= 0 (1 Š T )2( y1Š y2)=0in X .ThisconditionmatchestheonefoundinExample4.2.2 64

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forthequandles X consideredtherein.ThesecondstatementofTheorem6.1.1implies,inparticular,thatthecocycleinvariantisnon-trivialfor X =q/ (1 Š T )2and E =q/ (1 Š T )3,aswehaveseeninExample4.2.3.Notethatifweuse therearrangementofthearcs( w1,...,w6)wede“nedbeforeTheorem6.1.2,weget w1= y1,w2= y5,w3= y2,w4= y6,w5= y3,w6= y4withappropriatebasepoints. AswecanseefromFigure17, y1,y5arecolorsassignedtothecomponent K1, y2,y6arefor K2,and y3,y4arefor K3. z =( z1, ,z6) =( TŠ 1(1 Š T )2( z3Š z5) 0 ,TŠ 1(1 Š T )2( z5Š z1) 0 ,TŠ 1(1 Š T )2( z1Š z3) 0) where z1,z2belongto K1, z3,z4belongto K2,and z5,z6belongtocomponent K3. ByTheorem6.1.2,thenon-trivialcontributionto ( L )is ( t2 j =1 ( j) zj/ (1 Š T )2,t4 j =3 ( j) zj/ (1 Š T )2,t6 j =5 ( j) zj/ (1 Š T )2)=( tk,t,tŠ ( k + )) forsome k, ,where0 k, q Š 1,dependingonthevaluesof w3Š w5and w5Š w1. NotethatthismatchestheconclusionofExample4.2.3. 6.2ArelationtoAlexander-Conwaypolynomial Inthissectionwedescribearelationofcocycleinvariantsoriginatingfromextensions totheConwaypolynomial. LetL( T ) Z [ TŠ1 2,T1 2]bethe Conway-normalizedAlexanderpolynomial [30]. Inourcase,let ADLbethematrixobtainedfrom ADLbydeletingthe i thcolumn and i throwforsome i i =1 ,...,n ,let f ( T )=det( ADL) Z [ T1,TŠ 1]and and bethemaximalandminimaldegreeof f respectively.Then,L( T )= TŠ + 2f ( T ). 65

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The Conwaypolynomial L( z ) Z [ z ]isde“nedby L( TŠ1 2Š T1 2)=L( T ),where z = TŠ1 2Š T1 2. Proposition6.2.1 Lettheminimaldegreeof L( z ) bedenotedby min-deg L( z ) thenitsatises min-deg L( z ) m ,where m isthesmallestintegersuchthatthe cocycleinvariantdenedfromtheextensionof Zq[ T,TŠ 1] / (1 Š T )mto Zq[ T,TŠ 1] / (1 Š T )m +1isnon-trivial. Proof. Assumethat yA( X ) DL= 0and s ( y ) A( E ) DL= x = 0.Then y contributesanontrivialvaluetotheinvariant ( L )asinProposition6.1.1.Since x = 0thereexists i 1 i n ,suchthat xi =0.Let j beaninteger,1 j n ,with j = i .Let xbethe vector x withthe xjentrydeleted.Then,thereexists y = 0,where yisthevector y withthe j thentrydeleted,suchthat yA( X ) DL= 0.Thisimpliesthatdet A( X ) DL=0. Hence,det ADL 0(mod(1 Š T )m),andwehavemin-deg L( z ) m 66

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ABOUTTHEAUTHOR MarinaAppiouNikiforouwasbornonOctober9,1974inPlatres,Cyprus.She receivedaBachelorsDegreeinMathematicsfromUniversityofCyprusin1996.She earnedthefollowingdegreesfromUniversi tyofSouthFlorida:MastersDegreein Mathematicsin1998,andherDoctorofPhilosophyinMathematicswithemphasis inKnotTheory.Mrs.Nikiforoureceivedthreescholarshipsandworkedasateaching assistant.