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Polynomial quandle cocycles, their knot invariants and applications

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Title:
Polynomial quandle cocycles, their knot invariants and applications
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English
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Ameur, Kheira
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University of South Florida
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Subjects / Keywords:
Torus Knots
Twist knots
State-sum Invariants
Colorings
Tangle embedding
Dissertations, Academic -- Mathematics -- Doctoral -- USF   ( lcsh )
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bibliography   ( marcgt )
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ABSTRACT: A quandle is a set with a binary operation that satisfies three axioms that corresponds to the three Reidemeister moves on knot diagrams. Homology and cohomology theories of quandles were introduced in 1999 by Carter, Jelsovsky,Kamada, Langford, and Saito as a modification of the rack (co)homology theory defined by Fenn, Rourke, and Sanderson. Cocycles of the quandle (co)homology, along with quandle colorings of knot diagrams, were used to define a new invariant called the quandle cocycle invariants, defined in a state-sum form. This invariant is constructed using a finite quandle and a cocyle, and it has the advantage that it can distinguish some knots from their mirror images, and orientations of knotted surfaces. To compute the quandle cocycle invariant for a specific knot, we need to find a quandle that colors the given knot non-trivially, and find a cocycle of the quandle. It is not easy to find cocycles,since the cocycle conditions form a large, over-determined system of linear equations. At first the computations relied on cocycles found by computer calculations. We have seen significant progress in computations after Mochizuki discovered a family of 2- and 3-cocycles for dihedral and other linear Alexander quandles written by polynomial expressions. In this dissertation, following the method of the construction by Mochizuki, a variety of n-cocycles for n >̲ 2 are constructed for some Alexander quandles, given by polynomial expressions. As an application, these cocycles are used to compute the invariants for (2,n)-torus knots, twist knots and their r-twist spins. The calculations in the case of (2,n)-torus knots resulted in formulas that involved the derivative of the Alexander polynomial. Non-triviality of some quandle homology groups is also proved using these cocycles. Another application is given for tangle embeddings. The quandle cocycle invariants are used as obstructions to embedding tangles in links. The formulas for the cocycle invariants of tangles are obtained using polynomial cocycles, and by comparing the invariant values, information is obtained on which tangles do not embed in which knots. Tangles and knots in the tables are examined, and concrete examples are listed.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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by Kheira Ameur.
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Title from PDF of title page.
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Document formatted into pages; contains 89 pages.
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Includes vita.

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ABSTRACT: A quandle is a set with a binary operation that satisfies three axioms that corresponds to the three Reidemeister moves on knot diagrams. Homology and cohomology theories of quandles were introduced in 1999 by Carter, Jelsovsky,Kamada, Langford, and Saito as a modification of the rack (co)homology theory defined by Fenn, Rourke, and Sanderson. Cocycles of the quandle (co)homology, along with quandle colorings of knot diagrams, were used to define a new invariant called the quandle cocycle invariants, defined in a state-sum form. This invariant is constructed using a finite quandle and a cocyle, and it has the advantage that it can distinguish some knots from their mirror images, and orientations of knotted surfaces. To compute the quandle cocycle invariant for a specific knot, we need to find a quandle that colors the given knot non-trivially, and find a cocycle of the quandle. It is not easy to find cocycles,since the cocycle conditions form a large, over-determined system of linear equations. At first the computations relied on cocycles found by computer calculations. We have seen significant progress in computations after Mochizuki discovered a family of 2- and 3-cocycles for dihedral and other linear Alexander quandles written by polynomial expressions. In this dissertation, following the method of the construction by Mochizuki, a variety of n-cocycles for n > 2 are constructed for some Alexander quandles, given by polynomial expressions. As an application, these cocycles are used to compute the invariants for (2,n)-torus knots, twist knots and their r-twist spins. The calculations in the case of (2,n)-torus knots resulted in formulas that involved the derivative of the Alexander polynomial. Non-triviality of some quandle homology groups is also proved using these cocycles. Another application is given for tangle embeddings. The quandle cocycle invariants are used as obstructions to embedding tangles in links. The formulas for the cocycle invariants of tangles are obtained using polynomial cocycles, and by comparing the invariant values, information is obtained on which tangles do not embed in which knots. Tangles and knots in the tables are examined, and concrete examples are listed.
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Torus Knots.
Twist knots.
State-sum Invariants.
Colorings.
Tangle embedding.
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by KheiraAmeur Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:MasahikoSaito,Ph.D. BrianCurtin,Ph.D. MouradIsmail,Ph.D. NatashaJonoska,Ph.D. DavidRabson,Ph.D. DateofApproval:October30,2006 Keywords:Torusknots,Twistknots,State-suminvariants,Colorings,Tangleembedding. cCopyright2006,KheiraAmeur

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MywarmestappreciationsgotoProfessorsBrianCurtin,MouradIsmail,andNatashaJonoskaforacceptingtobepartofthesupervisingcommittee. IwouldalsoliketothankProfessorDavidRabsonofthePhysicsdepartmentforacceptingtobethechairpersonofthedefensecommittee. IamdeeplyindebtedtothemathematicsdepartmentoftheUniversityofSouthFloridaforitsgenerousnancial,mathematicalandpersonalsupport. Thankstoallthepeoplewhohavecontributedbytheirfriendshipandencouragementtotheadvancementofthiswork.

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Abstractv 1Introduction1 1.1Overview..................................1 1.2KnotDiagrams..............................3 1.3Quandles..................................3 1.4CohomologyTheoryofQuandles....................5 1.5ColoringsofKnotDiagramsbyQuandles................7 1.6TheQuandle2-CocycleInvariant....................8 1.7TheQuandle3-CocycleInvariant....................9 2PolynomialCocyclesofAlexanderQuandles11 2.1Polynomialn-Cocycles..........................11 2.2ExamplesofPolynomialCocycles....................15 3ColoringsandCocycleInvariantsofKnotswithAlexanderQuandles18 3.1ColoringsofKnotDiagramsbyAlexanderQuandles.........18 3.1.1Coloring(2;m)-TorusKnotsbyAlexanderQuandles.....18 3.1.2ColoringTwistKnotsbyAlexanderQuandles.........20 3.2KnotInvariantsby2-Cocycles......................21 3.2.1The2-cocycleInvariantsfor(2;n)-TorusKnots........21 3.2.2The2-cocycleInvariantsforTwistKnots............25i

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3.3.1The3-cocycleInvariantsfor(2;n)-TorusKnots........27 3.3.2The3-cocycleInvariantsforTwistKnots............31 4Applications36 4.1QuandleCocycleInvariantsforTwistSpunKnots...........37 4.1.1TwistSpinning..........................37 4.1.2Twist-spun(2;m)-TorusKnots.................40 4.1.3TwistSpunTwistKnots.....................42 4.1.4RemarksonNon-InvertibilityofTwistSpunKnots......50 4.2QuandleCocycleInvariantsandTangleEmbeddings..........53 4.2.1TanglesandTheirOperations..................53 4.2.2CocycleInvariantsasObstructions...............55 4.2.3TangleswithTwoTwists.....................57 4.2.4UsingTanglesandKnotTables.................66 4.2.5EmbeddingDisjointTangles...................77 ConcludingRemarks80 AppendixA.Non-trivialityof3-cocycleInvariantsforTwistKnots81 AppendixB.Valuesof3-cocycleInvariantsforTwistKnots83 References86 AbouttheAuthorEndPageii

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1.2Quandlerelationatacrossing......................7 1.3The2-cocycleconditionandtypeIIImove...............8 1.4Quandlecoloringsofregions.......................9 3.1TorusknotsK(2;m)...........................19 3.2Twistknots................................20 4.1Amovieoftwistspinning........................38 4.2Apartofadiagramoftwist-spuntrefoil................39 4.3Additionoftangles............................54 4.4Closures(numeratorN(T),denominatorD(T))oftangles......54 4.5Somerationaltangles..........................55 4.6Tanglewithtwotwists..........................57 4.7Twostringswithanantiparallelorientation,caseI..........59 4.8Twostringswithanantiparallelorientation,caseII..........60 4.9Case1...................................61 4.10Case2...................................63 4.11Case3...................................64 4.12Tableoftangles.............................66 4.13Tangle717,NWin,SWout.......................70 4.14Tangle717,NWin,SWin........................71 4.15Tangle76.................................73 4.16Tangle77.................................74iii

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4.18Tangle718.................................76iv

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Tocomputethequandlecocycleinvariantforaspecicknot,weneedtondaquandlethatcolorsthegivenknotnon-trivially,andndacocycleofthequandle.Itisnoteasytondcocycles,sincethecocycleconditionsformalarge,over-determinedsystemoflinearequations.Atrstthecomputationsreliedoncocyclesfoundbycom-putercalculations.WehaveseensignicantprogressincomputationsafterMochizukidiscoveredafamilyof2-and3-cocyclesfordihedralandotherlinearAlexanderquan-dleswrittenbypolynomialexpressions. Inthisdissertation,followingthemethodoftheconstructionbyMochizuki,avarietyofn-cocyclesforn2areconstructedforsomeAlexanderquandles,givenbypolynomialexpressions.Asanapplication,thesecocyclesareusedtocomputetheinvariantsfor(2;n)-torusknots,twistknotsandtheirr-twistspins.Thecalculationsv

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Anotherapplicationisgivenfortangleembeddings.Thequandlecocycleinvariantsareusedasobstructionstoembeddingtanglesinlinks.Theformulasforthecocycleinvariantsoftanglesareobtainedusingpolynomialcocycles,andbycomparingtheinvariantvalues,informationisobtainedonwhichtanglesdonotembedinwhichknots.Tanglesandknotsinthetablesareexamined,andconcreteexamplesarelisted.vi

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Thefundamentalquandleisdenedsimilarlyforknottedsurfacesin4-spaceusingdiagrams[13].Althoughfundamentalknotquandlesarestronginvariants,itisnoteasytousethemtodistinguishknotsbydirectcalculations,sincetheyaredenedbygeneratorsandrelations.Onewaytousethemistocalculatethenumberofhomeomorphismstoagivennitequandle.Forexample,Fox3-coloringsarehomeo-morphismsofthefundamentalknotquandletothedihedralquandleoforder3. Inthelate1999'sandearly2000's,homologyandcohomologytheoriesforquandlesappearedin[9],asamodicationoftherack(co)homologytheorydenedin[19].Cocyclesofthequandlecohomology,alongwithquandlecoloringsofknotdiagrams,wereusedtodeneanewinvariantcalledthequandlecocycleinvariant[9].The1

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Tocomputethequandlecocycleinvariantforaspecicknot,weneedtondaquandlethatcolorsagivenknotnon-trivially,andtondacocycleofthequan-dle.Itisnoteasytondcocycles,sincethecocycleconditionsformalarge,over-determinedsystemoflinearequations.Atrstthecomputationsreliedoncocyclesfoundbycomputercalculations.WehaveseensignicantprogressincomputationsafterMochizuki[31]discoveredafamilyof2-and3-cocyclesfordihedralandotherlinearAlexanderquandleswrittenbypolynomialexpressions.Formulasforimportantfamiliesofknotsandknottedsurfacesandtheirapplicationsfollowed[1,24]. Inknottheory,wheneveranewinvariantisdened,computationoftheinvariantforknotsintheknottable,orforsometypicalfamiliesofknotsareperformed,sothatwecangetsomedatatobeusedfordistinguishingknotsandforotherapplications.Thequandlecocycleinvariantsdenealargefamilyofknotinvariants.Computationoftheseinvariantsinvolvendingcocycles.Oneofthemaingoalsofthisdissertationistodevelopnewcomputationsofthequandlecocycleinvariants,andthisisdonebyconstructinganewfamilyofcocycles.Thisdissertationcontainsfourchapters.Inthecurrentchapter,Chapter1,werecallstandarddenitions.InChapter2wegiveanewconstructionofalargefamilyofn)]TJ/F1 11.95 Tf 9.3 0 TD[(cocyclesforsomeAlexanderquandles.InChapter3wecomputecoloringsbyAlexanderquandlesfortwofamiliesofknots,(2;n)-torusknotsandtwistknots,thenwecomputethe2-and3-cocycleinvariantsfortheseknotsusingthecocyclesconstructedinchapter2.Inthecaseof(2;n)-torusknots,theformulasobtainedinvolvethederivativeoftheAlexanderPolynomial.Theseformulasare,then,usedtoprovenontrivialityofsomequandle(co)homologygroups.Chapter4containstwoapplications;rstwecomputethequandlecocycleinvariantforther-twistspinoftorusknotsandtwistknots,then,cocycleinvariantsareusedasobstructionstoembeddingtanglesinlinks.Theformulasforthecocycleinvariantsoftanglesareobtainedusingpolynomialcocycles,andbycomparingtheinvariantvalues,someinformationisobtainedonwhichtanglesdonotembedinwhichknots.Tanglesandknotsinthetablesareexamined,andconcreteexamples2

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Thefollowingsectionsconsistofreviewsofstandarddenitionsthatareusedthroughoutthisdissertation.1.2KnotDiagrams

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(I)Foranya2X,aa=a. (II)Foranya;b2X,thereisauniquec2Xsuchthata=cb. (III)Foranya;b;c2X,wehave(ab)c=(ac)(bc): Thefollowingtypicalexamplesarefoundintheliteraturementionedabove.Example1.3.2

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LetCRn(X)bethefreeAbeliangroupgeneratedbyn-tuples(x1;:::;xn)ofelementsofaquandleX.Deneahomomorphism@n:CRn(X))166(!CRn)]TJ/F6 7.97 Tf 6.58 0 TD[(1(X)by:@n(x1;:::;xn)=nXi=2()]TJ/F1 11.95 Tf 9.29 0 TD[(1)i[(x1;x2;:::;xi)]TJ/F6 7.97 Tf 6.58 0 TD[(1;xi+1;:::;xn))]TJ/F1 11.95 Tf 19.26 0 TD[((x1xi;x2xi;:::;xi)]TJ/F6 7.97 Tf 6.58 0 TD[(1xi;xi+1;:::;xn)] forn2and@n=0forn1.ThenCR(X)=fCRn(X);@ngisachaincomplex.LetCDn(X)bethesubsetofCRn(X)generatedbyn-tuples(x1;:::;xn)withxi=xi+1forsomei2f1;:::;n)]TJ/F1 11.95 Tf 13.15 0 TD[(1gifn2;otherwiseletCDn(X)=0:IfXisaquandle,then@n(CDn(x))CDn)]TJ/F6 7.97 Tf 6.59 0 TD[(1(X)andCD(X)=fCDn(X);@ngisasubcomplexofCR(X).PutCQn=CRn(X)=CDn(X)andCQ(X)=fCQn(X);@0ngwhere,@0nistheinducedhomomorphism.Henceforth,allboundarymapsmaybedenotedby@n.ThesuperscriptsR;QandD,respectively,representrack,quandle,anddegeneratechaincomplexes.ForanAbeliangroupG,denethechainandthecochaincomplexesbyCW(X;G)=CW(X)G,@=@id;CW(X;G)=Hom(CW(X);G),=Hom(@;id),intheusualway,whereW=D;R;Q:5

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wheref2CnW(X;G).Amapf2Cn+1W(X;G)iscalledacoboundary,ifthereisamapg2CnW(X;G)suchthatng=f. Thegroupsofcyclesandboundariesaredenotedrespectivelybyker(@)=ZWn(X;G)CWn(X;G)andIm(@)=BnW(X;G)CnW(X;G)whilethecocyclesandcoboundariesaredenotedrespectivelybyker()=ZnW(X;G)CnW(X;G)andIm()=BnW(X;G)CnW(X;G).Then-thquandlehomologygroupwithcoecientgroupGisdenedbyHQn(X;G)=Hn(CQ(X;G))=ZQn(X;G)=BQn(X;G): forallx;y;z2X,and(x;x)=0forallx2X.A3-cocycleisregardedasafunction:XXX!Awiththe3-cocyclecondition(x;z;w)+(x;y;z)+(xz;yz;w)=(xz;y;w)+(x;y;w)+(xw;yw;zw)6

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LetXbeaxedquandle.LetKbeagivenorientedclassicalknotorlinkdiagram,andletRbethesetof(over-)arcs.Thenormals(normalvectors)aregiveninsuchawaythattheorderedpair(tangent,normal)agreeswiththeorientationoftheplane,seeFig.1.2.A(quandle)coloringCisamapC:R!Xsuchthatateverycrossing,therelationdepictedinFig.1.2holds.Specically,letbetheover-arcatacrossing,andletandbetheunderarcs,suchthatthenormaloftheover-arcpointsfromto,thenC()C()=C()holds. Theelementa=C()2Xassignedtoanarciscalledthecolorof.The(ordered)colors(C(),C())arecalledsourcecolors.LetColX(D)denotethesetofcoloringsofaknotdiagramDofaknotKbyaquandleX.

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iscalledthequandlecocycleinvariant.

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ForacoloringC,thereisacoloringofregionsthatextendCasdepictedinFig.1.4.Let(x;y;z)=(x;y;z)bethecolorsnearacrossingsuchthatxisthecoloroftheregion(calledthesourceregion)fromwhichbothorientationnormalsofover-andunder-arcspoint,yisthecoloroftheunder-arc(calledthesourceunder-arc)fromwhichthenormaloftheover-arcpoints,andzisthecoloroftheover-arc.SeeFig.1.4.

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SomepolynomialcocycleshavebeenconsideredbyMochizuki[31].Ourgoalistoexaminecertainpolynomialsinfullgenerality,ndingwhichpolynomialsbecomecocyclesforwhichAlexanderquandles.2.1Polynomialn-Cocycles Ifan=0,Thenfisann-cocycle(2ZnQ(X;A)).2. Ifan=pmn(forapositiveintegermn),thenfisann-cocycleifg(t)divides1)]TJ/F4 11.95 Tf 11.95 0 TD[(ta,wherea=a1+a2++an)]TJ/F6 7.97 Tf 6.58 0 TD[(1+an.11

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Proof.Letl(t)betheinverseof1)]TJ/F4 11.95 Tf 11.95 0 TD[(ta1+a2+:::+aninX,andletf1(x1;x2;:::;xn)=()]TJ/F1 11.95 Tf 9.3 0 TD[(1)n+1l(t)(x1)]TJ/F4 11.95 Tf 11.95 0 TD[(x2)a1(x2)]TJ/F4 11.95 Tf 11.95 0 TD[(x3)a2(xn)]TJ/F6 7.97 Tf 6.59 0 TD[(1)]TJ/F4 11.95 Tf 11.96 0 TD[(xn)an)]TJ/F12 5.98 Tf 5.75 0 TD[(1xann; Proof.SinceX=Rp,andnisodd,wehave1)]TJ/F4 11.95 Tf 11.18 0 TD[(ta=2inX.Then1)]TJ/F4 11.95 Tf 11.18 0 TD[(taisinvertibleinX.SobyCorollary2.1.2,fisacoboundary.214

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Proof.InX,1)]TJ/F4 11.95 Tf 12.32 0 TD[(ta1+a2++an=)]TJ/F4 11.95 Tf 9.3 0 TD[(tk,then1)]TJ/F4 11.95 Tf 12.32 0 TD[(ta1+a2;++anisinvertible,thenfisacoboundarybyCorollary2.1.2.2Example2.1.5 MorespecicpolynomialsandAlexanderquandlesweconsiderareasfollows.

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AlexanderquandleZ2[t;t)]TJ/F6 7.97 Tf 6.58 0 TD[(1]=(t2+t+1)hasanon-trivial2-cocyclef(x;y)=(x)]TJ/F4 11.95 Tf 12.45 0 TD[(y)2y.This4elementquandleiswell-known.Itisiso-morphictothequandleconsistingof120degreerotationsofaregulartetrahedron.Itisknowntohave2-dimensionalcohomologygroupZ2withZ2coecient.Theinvariantvaluesarealloftheform16,ork[4+12u(t+1)],soweconjecturethatitisalwaysthecase.Itisalsoaninterestingproblemtocharacterizethevaluesofthisinvariant. Thequandlemustbemodg(t)whereg(t)dividest5)]TJ/F1 11.95 Tf 12.69 0 TD[(1,butt5)]TJ/F1 11.95 Tf 12.69 0 TD[(1isfactoredintoprimepolynomials(t+1)(t4+t3+t2+t2+1)mod2,sowesetg(t)=t4+t3+t2+t2+1.TheAlexanderquandleweuseinthiscase,thus,isZ2[t;t)]TJ/F6 7.97 Tf 6.58 0 TD[(1]=(t4+t3+t2+t+1)with2-cocycle:f(x;y)=(x)]TJ/F4 11.95 Tf 12.9 0 TD[(y)22y,whichgivesnon-trivialinvariantsbycomputercalculations[41]. Wefactort9)]TJ/F1 11.95 Tf 9.69 0 TD[(1mod2to(t+1)(t2+t+1)(t6+t3+1),sowetryAlexanderquandleZ2[t;t)]TJ/F6 7.97 Tf 6.58 0 TD[(1]=(t6+t3+1),whichgivesnon-trivialinvariantsbycomputercalculations[41].(b)

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Manycomputercalculationsin[41]arebasedonthesecocycles.17

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Proof.Letmbeapositiveinteger.Notethat1)]TJ/F4 11.95 Tf 10.67 0 TD[(tk)]TJ/F6 7.97 Tf 6.59 0 TD[(1=k.Thenthislemmafollowsfromtheinductionusingthecalculationt(tk)]TJ/F6 7.97 Tf 6.58 0 TD[(1a+kb)+(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)(tka+k+1b)=(t2k)]TJ/F6 7.97 Tf 6.59 0 TD[(1+t(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)k)a+(tk+(1)]TJ/F4 11.95 Tf 11.96 0 TD[(t)k+1)b=t(1)]TJ/F4 11.95 Tf 11.95 0 TD[(tk)+t(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)k)a+((1)]TJ/F4 11.95 Tf 11.95 0 TD[(k+1)+(1)]TJ/F4 11.95 Tf 11.96 0 TD[(t)k+1)b=tk+1a+k+2b:2Corollary3.1.2 Proof.InLemma3.1.1,thetopandthebottomcolorvectorscoincideifandonlyif(a;b)=(tm)]TJ/F6 7.97 Tf 6.58 0 TD[(1a+mb;tma+m+1b)19

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Proof.Theendcolors(thecolorsofthetopandbottomrightarcs)oftheantiparallelstringsare,respectively,a0=a+n(1)]TJ/F4 11.95 Tf 12.78 0 TD[(t)(b)]TJ/F4 11.95 Tf 12.79 0 TD[(a)andb0=b+n(1)]TJ/F4 11.95 Tf 12.78 0 TD[(t)(b)]TJ/F4 11.95 Tf 12.79 0 TD[(a)asdepictedinthegure.Thetwocrossingsbelowantiparallel2ncrossingsgivethe20

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Proof.Itisknownthatifacocycleisacoboundary,thenthecocycleinvariantistrivial(beingtrivialmeansthattheinvariantvalueisapositiveintegercorrespondingtothenumberofcoloringsinastate-sumform,orthatnumberofcopiesofzerosinamulti-setform)[9].Itis,therefore,sucienttoshowthatthereisacoloringcontributinganon-trivialvalue. (i)ForX=Z2[t;t)]TJ/F6 7.97 Tf 6.59 0 TD[(1]=2n+1,letf(x1;x2)=(x1)]TJ/F4 11.95 Tf 11.78 0 TD[(x2)2nx2inLemma2.1.1,sothata1=2nanda2=1.Notethat1)]TJ/F4 11.95 Tf 12.66 0 TD[(t(a1+a2)=(1)]TJ/F4 11.95 Tf 12.65 0 TD[(t)2n+1.Take(1;0)2XXasatopcolorvector,whichextendstoacoloringofK(2;m),wherem=2n+23

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(ii)Inthiscasetakea1=pnanda2=1asbefore,then1)]TJ/F4 11.95 Tf 12.96 0 TD[(t(a1+a2)=(1)]TJ/F4 11.95 Tf -423.96 -20.91 TD[(t(pn+1)=2)(1+t(pn+1)=2).If(pn+1)=2isodd,then(pn+1)=2divides(1+t(pn+1)=2),andifeven,itdivides(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t(pn+1)=2),hencetheresultfollowsbythesameargument.2Proposition3.2.3 Ifmisanegativeinteger,thentheinvariantconsistsofthenegativesofthein-variantforK(2;jmj). Proof.ByLemma2.1.1,indeedf2Z2Q(X;A).Byassumptionanytopcolorvector(a;b)extendstoacoloring.Lets=a)]TJ/F4 11.95 Tf 11.95 0 TD[(b.ThenbyLemma3.2.1wehavef(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)(a1+a2)j(a;b)2XXg=ftjXjs(a1+a2)js2Xg Ifmisnegative,thenallcrossingsarenegative.ThenconsiderthediagramofK(2;m)thatisthemirrorofthediagramusedaboveofK(2;jmj),withoppositeorientation.ThenalsoconsiderthecolorsofK(2;m)atthebottomarcs(a;b).Thenthecontributionfromthecoloringinducedbythisbottomcolorvectorcoincideswiththenegativeoftheoriginal.Hencetheinvariantf(K(2;m))isthemultisetthatconsistsofthenegativeoff(K(2;jmj)).2Example3.2.4

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Proof.Takea1=1anda2=pnforapositiveintegern,thenthetopcolorvector(1;0)givesanon-trivialvalueinLemma3.3.1.Theconditionthatmdivides1)]TJ/F4 11.95 Tf 12 0 TD[(ta1+a2iscomputedasinCorollary3.2.2.2Example3.3.3

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Proof.WecomputemXk=1a1k()]TJ/F4 11.95 Tf 9.29 0 TD[(t)ka2=mXk=1(1)]TJ/F4 11.95 Tf 11.95 0 TD[(tk)]TJ/F6 7.97 Tf 6.58 0 TD[(1)a1()]TJ/F4 11.95 Tf 9.3 0 TD[(t)ka229

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()]TJ/F4 11.95 Tf 9.29 0 TD[(tmm+1)=(1)]TJ/F4 11.95 Tf 11.96 0 TD[(t)=[()]TJ/F4 11.95 Tf 9.29 0 TD[(tm)=(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)](1)]TJ/F4 11.95 Tf 11.95 0 TD[(tm):

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Weobtainednon-trivialvaluesforthefollowingp,2
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HencewehavethefollowingcorollaryCorollary3.3.9 Belowwegiveasummaryofcomputationalresultsforotherprimes.First,weusedcocyclesoftheformf(x;y;z)=(x)]TJ/F4 11.95 Tf 12.16 0 TD[(y)a1(y)]TJ/F4 11.95 Tf 12.17 0 TD[(z)a2wherea1anda2are1orp.Wealwaysobtainedtrivialinvariantsforthecocyclesf(x;y;z)=(x)]TJ/F4 11.95 Tf 11.94 0 TD[(y)(y)]TJ/F4 11.95 Tf 11.94 0 TD[(z)andf(x;y;z)=(x)]TJ/F4 11.95 Tf 12.13 0 TD[(y)p(y)]TJ/F4 11.95 Tf 12.13 0 TD[(z)p,sowelisttheothercocycles,forwhichtheinvariantisnon-trivial.

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Non-invertibilityofthe2-twist-spuntrefoilwastherstapplicationofthequandlecocycleinvariants[6].Twodevelopmentsmadeitpossible,sincethen,toextendthisresulttoalargenumberoftwist-spunknots;simplerdiagramsoftwist-spunknotsgivenin[1]andanexplicitformulaof3-cocyclesofdihedralquandlesRpdiscoveredbyMochizuki[31].Itisofinterest,then,toobtainformulasofquandlecocycleinvariantsfortwist-spunknotsusingourpolynomialcocyclestoseewhichtwist-spunshavenon-trivialinvariantswithsomeAlexanderquandles,aswellastoseewhethertheyprovideobstructionstonon-invertibility.Intherstsectionweinvestigatetheseproblems. Inthesecondsectionofthischapter,weusecocycleinvariantsasobstructionstoembeddingtanglesinknotsandlinks.InapplicationofknottheorytoenzymeactionsonDNA[17],tangleequationswereformulatedandsolved,thatweresetupusingexperimentalresults.Intangleequations,whichtanglescanbeembeddedinagivenknotisapartoftheproblem.ThetangleembeddingproblemwaslaterstudiedbyKrebes[27]usingevaluationsoftheJonespolynomial,thataredeterminantsoftheclosuresofagiventangle.Thismethodwasinterpretedintermsofhomologyofdoublebranchedcoversalongknots[35],andthentheresultswerefurthergeneralizedindierentdirections[15,33].Weusecocycleinvariantsasobstructionstotangleembeddings.Firstwedenecocycleinvariantsfortanglesbyrequiringthatendpoints36

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InFig.4.2,eachverticalcrosssectionofthebrokensurfacediagrammarkedby(1)through(5)correspondstothediagraminatwistdepictedinFig.4.1alsomarkedbythecorrespondingnumbers(1)through(5).SmallblackdotsinFig.4.2between(1)and(2),(4)and(5),respectively,arebranchpointsoftheprojectedsurfacecorrespondingtothetypeIReidemeistermovethatoccurbetween(1)and(2),(4)and(5)inFig.4.1. Inparticular,thediagramofTcanberegardedasapart(slice)ofthediagramDrT,andthissituationisrepresentedbytheinclusionmapi:T!DrT.LetColX(T)andColX(DrT)bethesetofcoloringsofTandDrT,respectively,byaquandleX.Leti:T!DrTbetheinclusionmap,andleti:ColX(DrT)!ColX(T)etheinducedmap.PutColrX(T)=Im(i)2ColX(T).AsamiandSatoh[1]provedthatthemapiisinjective,andC2ColX(T)belongstoColrX(T)ifandonlyifCar)]TJ/F1 11.95 Tf 10.4 2.95 TD[(=Cwherea)]TJ/F1 11.95 Tf 10.04 1.79 TD[(isthecoloroftheterminalarcofT.HerethenotationCaforC2ColX(T)38

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AsbeforeweconsiderthecaseX=Zp[t;t)]TJ/F6 7.97 Tf 6.59 0 TD[(1]=h(t)andA=Zp[t;t)]TJ/F6 7.97 Tf 6.59 0 TD[(1]=g(t),h(t);g(t)2Zp[t;t)]TJ/F6 7.97 Tf 6.59 0 TD[(1]suchthatg(t)dividesh(t)andpisaprime.Letf:X3!Abedenedbyf(x1;x2;x3)=(x1)]TJ/F4 11.95 Tf 11.63 0 TD[(x2)a1(x2)]TJ/F4 11.95 Tf 11.63 0 TD[(x3)a2whereai=pmifori=1;2,wheremisarenon-negativeintegers.ThenbyLemma2.1.1,f2Z3Q(X;A).Lemma4.1.1

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Proof.Recallthatanr-twistspunknotiscolorednon-triviallybyanAlexanderquandleX,wehavetohave1)]TJ/F4 11.95 Tf 12.95 0 TD[(tr=0inX.Sincetrivialcoloringsgivetrivialcontributions,wemayassumenowthat1)]TJ/F4 11.95 Tf 10.74 0 TD[(tr=0inX.Notealsothatallthreecasesimplyniscoprimewithp. 2.Case(a)ta1+a2=()]TJ/F1 11.95 Tf 9.29 0 TD[(1)a1+a2 2=1.Thecontributiontothestate-sumofacoloringinducedbya;bisgivenbyr[)]TJ/F4 11.95 Tf 9.3 0 TD[(nt)]TJ/F8 7.97 Tf 6.59 0 TD[(a1+(1+n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t))a1+a2](a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2)]TJ/F1 11.95 Tf 9.3 0 TD[(()]TJ/F4 11.95 Tf 9.3 0 TD[(nta2+[1+n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)]a1+a2)r)]TJ/F6 7.97 Tf 6.58 0 TD[(1Xk=0t(a1+a2)k](a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2: 2k=r)]TJ/F6 7.97 Tf 6.59 0 TD[(1Xk=01=r:

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2=)]TJ/F1 11.95 Tf 9.3 0 TD[(1.Sincerisamultipleof4,wehaver)]TJ/F6 7.97 Tf 6.58 0 TD[(1Xk=0t(a1+a2)k=r)]TJ/F6 7.97 Tf 6.59 0 TD[(1Xk=0()]TJ/F1 11.95 Tf 9.29 0 TD[(1)a1+a2 2k=r)]TJ/F6 7.97 Tf 6.58 0 TD[(1Xk=0()]TJ/F1 11.95 Tf 9.3 0 TD[(1)k=0:

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Nowsupposethatp6=3,thenwehavetp=t,ortp=)]TJ/F4 11.95 Tf 9.3 0 TD[(t)]TJ/F1 11.95 Tf 12.16 0 TD[(1.Iftp=t,thenwehavetpm=t,sota1+a2=t2,andr)]TJ/F6 7.97 Tf 6.59 0 TD[(1Xk=0t(a1+a2)k=r)]TJ/F6 7.97 Tf 6.58 0 TD[(1Xk=0t2k=(1+t2+t4)(r=3)=0.Thenthecontributionisequaltor(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.29 0 TD[(nt)]TJ/F8 7.97 Tf 6.58 0 TD[(a1+(1+n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t))a1+a2]:

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Ifm1,m2arebotheven,thenta1+a2=t2hencePr)]TJ/F6 7.97 Tf 6.59 0 TD[(1k=0=t(a1+a2)k=(1+t2+t4)(r=3)=0,andthecontributionisequaltor(a)]TJ/F4 11.95 Tf 11.96 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.3 0 TD[(nt)]TJ/F8 7.97 Tf 6.58 0 TD[(a1+(1+n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t))a1+a2]=r(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.3 0 TD[(nt)]TJ/F6 7.97 Tf 6.58 0 TD[(1+(1+n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t))2]=r(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.3 0 TD[(n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)+(1+2n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)+n2(1)]TJ/F4 11.95 Tf 11.96 0 TD[(t)2]=r(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.3 0 TD[(n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)+1+2n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)+nt]49

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Ifm1,m2arebothodd,thenta1+a2=(1)]TJ/F4 11.95 Tf 12.9 0 TD[(t)2=)]TJ/F4 11.95 Tf 9.3 0 TD[(t,hencePr)]TJ/F6 7.97 Tf 6.59 0 TD[(1k=0=t(a1+a2)k=Pr)]TJ/F6 7.97 Tf 6.58 0 TD[(1k=0()]TJ/F4 11.95 Tf 9.3 0 TD[(t)k=0,andthecontributionisgivenbyr(a)]TJ/F4 11.95 Tf 11.96 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.3 0 TD[(nt)]TJ/F8 7.97 Tf 6.58 0 TD[(a1+(1+n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t))a1+a2]=r(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.3 0 TD[(nt+1+2n(1)]TJ/F4 11.95 Tf 11.96 0 TD[(ta1)+(n2(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)2)a1]=r(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)a1+a2[)]TJ/F4 11.95 Tf 9.3 0 TD[(nt+1+2nt+n(1)]TJ/F4 11.95 Tf 11.95 0 TD[(t)]=024.1.4RemarksonNon-InvertibilityofTwistSpunKnots LetMbeamulti-setofelementsofanabeliangroupA.Let f(rK). Proof.LetFbeaknottedsurface,)]TJ/F4 11.95 Tf 9.3 0 TD[(Fitsorientationreversedcounterpart,andletFbethemirrorimageofF.Thenitisknown[10]thatforaknottedsurfaceF,thecocycleinvariantsatisesf()]TJ/F4 11.95 Tf 9.3 0 TD[(F)= f(F). Ontheotherhand,Litherland[28]provedthatifaclassicalknotKisinvertible,thenitstwist-spunrKisamphicheiral,sothatFisequivalenttoF.Hencewehavef()]TJ/F4 11.95 Tf 9.3 0 TD[(rK)=f()]TJ/F1 11.95 Tf 9.3 0 TD[((rK))= f(rK).2

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Notethatthecocycleinvariantsinvestigatedinthissectionhavetheformf(DrK)=fgsa1+a2js2Xg; Forthecaseoftorusknots,althoughitisprovedin[21]thatalltwistspuntorusknotsarenon-invertible,theargumentappliesonlytothesphericalcase.Ifthenon-invertibilityisdetectedbyquandlecocycleinvariants,ontheotherhand,thenon-invertibilityremainsvalidforhighergenussurfacesobtainedbyaddingtrivial1-handles. Usinglemma4.1.1andaMapleprogram,weobtainedthenon-invertibiltyofther-twistK(2;9)torusknot,whererisamultipleof18,byconrmingthattheinvariantvaluesarenotsymmetricintheabovesense.Thiscaseisnotincludedin[1]inwhichtheyuseddihedralquandles.OurMapleprogramcouldnotnishcomputationsforK(2;m)whenm>11.Wesummarizetheresultasthefollowingproposition.Proposition4.1.7 Proof.TakethequandleX=A=Zp[t;t)]TJ/F6 7.97 Tf 6.58 0 TD[(1]=9(t)andthecocyclef(x1;x2;x3)=(x1)]TJ/F4 11.95 Tf -423.96 -20.91 TD[(x2)(x1)]TJ/F4 11.95 Tf 11.95 0 TD[(x2)3.UsingourMapleprogram,weobtainthattheinvariantf(rK(2;9))is94143178827+188286357654U(2t7+2t6+t4+t3+2t+2),whererisamultipleof18.Then51

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Inthissectionweusequandlecocycleinvariantsasobstructionstoembeddingtanglesinknots.Weareinterestedintwotypesoftangles:tanglesthatareadditionoftworationaltangleswithmandnverticaltwistsrespectively,andtanglesupto7crossingsthatwereclassiedin[26].Thelatteronesareparametrizedbyapairofnumbersinasymbolsimilartothoserepresentingknots.Thesectionisorganizedasfollows.Firstwereviewdenitions,thenstateandproveatheoremthatisusedasobstructionstotangleembeddings.Inthethirdsubsection,wecomputeanduseinvariantsoftanglesthatareobtainedbytangleadditionsoftwoparallelorantiparallelstrings.Inthelastsubsection,weinvestigatetangleembeddingsforthetableoftanglesandthetableofknots.4.2.1TanglesandTheirOperations An(n;n)-tangleiscommonlyabbreviatedbyann-tangle.Inthisdissertationwearemostlyinterestedin2-tangles,andforsimplicityweshallrefertothemasjusttangles.WewillcallthefourpointsofthetanglethatlieontheboundaryoftheballB3,NE,NW,SE,SW(wheretheabbreviationsreferto,northeast,northwestetcetera).ThesepointscanbepreciselydescribedinR3intermsofthefollowing53

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TheadditionT1+T2oftwotanglesT1,T2isanothertangledenedfromtheoriginaltwoasdepictedinFig.4.3.Theclosuresaretwomethodsofobtainingalinkfromatanglebyclosingtheendpoints,andtherearetwowayscalledthenumeratorN(T)anddenominatorD(T)ofatangleT,denedasdepictedinFig.4.4.Thesedenitionscanbefound,forexample,in[32]. Figure4.4:Closures(numeratorN(T),denominatorD(T))oftangles

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ForacoloringCofatanglediagramT,aregioncoloringsaredenedinasimilarmannerasintheknotcase.Inthiscase,weallowregioncolorstochange(notnecessarilycoloredbythesameelementastheoneassignedtotheboundarypoints).55

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LetfTi:i=1;:::kgbeanitefamilyofdisjointtanglediagrams.Thentheinvariant(fTi:i=1;:::kg)isdenedinamannersimilartotheabove.Speci-cally,foreachcoloringthatismonochromaticontheboundarypointsofalltanglesinthefamily,denethecontributiontobethesumofcontributionsfromeachtanglewithrespecttothiscoloring.Thenthecocycleinvariantisdenedasamulti-setwithrespecttoallcolorings. Thequandle2-and3-cocycleinvariantsaredenedfortanglesinamannersim-ilartotheknotcase,anddenotedby(T).Let@TdenotetheboundarypointsofagiventangleT.Let(T;x)=XC2Colx(T)YB(C;)whereColx(T)=fC2ColX(T)jC(@T)=xgforx2X.Then(T)=Xx2X(T;x). LetM,Nbetwomultisets.Theinclusionofmultisetsisdenotedbym,andisdenedasfollows:ifanelementxisrepeatedntimesinamultiset,callnthemultiplicityofx.ThenMmNmeansthatifx2M,thenx2N,andthemultiplicityofxinMislessthanorequaltothemultiplicityofxinN.Theorem4.2.2 SupposeanitesetoftanglesT1;:::;TkembedsdisjointlyinalinkL,forapositiveintegerk.Thenwehave(fTi:i=1;:::;kg)m(L). Proof.SupposeadiagramofTembedsinadiagramofL.WecontinuetouseTandLforthesediagrams.ForacoloringCofT,letxbethecoloroftheboundarypoints.ThenthereisauniquecoloringC0ofLsuchthattherestrictionofC0onTisCandallthearcsofLoutsideofTreceivethecolorx.Thenthecontributionof56

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Wemayconsideronlythesethreecasesforthefollowingreasons.First,werestrictourselvesheretotanglesthatdonothaveaseparatecomponentofaclosedcircleinside.Thisrestrictionexcludesthecasewhenmandnarebotheven.Ifbothtwistsareparallel,thenitisCase3,andtheorientationsofmiddlearcs(thearcsthatconnecttwotwistparts)aredeterminedbytherequirementthatbothbeparallel.NotethatiftheorientationoftheNWisinward,thenthatofNEisoutward.HencetherotationofthistangleaboutthehorizontalaxismaketheorientationofNWoutward,sothatwecanonlyconsiderthecasewhereNWisinward,inthiscase. Thesameapplieswhenbothareantiparallel,givingrisetotheCase2.Therequirementthatbothantiparallel,inthiscase,determinestheorientationsofthemiddlearcs.Thentheparitiesoftwistsaredeterminedfromtheseorientationtobebothodd.ThisistheCase2. Thenthecasewithoneparallel,oneantiparallelremains.IfthelefttangleisparallelasdepictedinFig.4.9,andNWisinward,thenthemiddlearcistobeorientedasdepicted,andthen,therighttanglemusthaveeventwists.Theonlyothercaseisthesametypewiththeoppositeorientation.Theinvariant,inthecasewithoppositeorientationsisthenegativeoftherstcase,soweonlyconsiderthecaseofNWinward.Thisexhauststheabovecases. Inthissectionwecomputethe3-cocycleinvariantforthetanglesofCase1,Case2,andCase3,withthecocyclef(x1;x2;x3)=(x1)]TJ/F4 11.95 Tf 11.06 0 TD[(x2)a1(x2)]TJ/F4 11.95 Tf 11.06 0 TD[(x3)a2,wherea1=pm1,a2=pm2.Forthispurposewecomputethecontributionfromantiparallelstringstotheinvariant. FirstwecomputethecontributionfromtwoantiparallelstringsdepictedinFig.4.758

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InallwhatfollowsassumethataisthecoloroftheNWarc,andbisthecolorofthearcbetweentheNWarcandNEarc,unlessotherwisespecied.

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Proof.Assigncolorsa,bfortheNWarcandthearcbetweenNWandNEarcs.Forthelefttwists,ifmispositive,thenwehavethetwistsdepictedinthebottomofFig.4.8.Inparticular,theSWarcandthearcbetweenSWandSEarecoloredbyb+m0(1)]TJ/F4 11.95 Tf 12.63 0 TD[(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)(b)]TJ/F4 11.95 Tf 12.63 0 TD[(a)andt)]TJ/F6 7.97 Tf 6.58 0 TD[(1a+(1)]TJ/F4 11.95 Tf 12.63 0 TD[(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)b+m0(1)]TJ/F4 11.95 Tf 12.63 0 TD[(t)]TJ/F6 7.97 Tf 6.59 0 TD[(1)(b)]TJ/F4 11.95 Tf 12.63 0 TD[(a),respectively,wherem=2m0+1.Wemusthaveb+m0(1)]TJ/F4 11.95 Tf 12.77 0 TD[(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)(b)]TJ/F4 11.95 Tf 12.77 0 TD[(a)=a.Tosimplifythecalculations,wetakethecondition1+m0(1)]TJ/F4 11.95 Tf 11.87 0 TD[(t)]TJ/F6 7.97 Tf 6.59 0 TD[(1)=0.Hencem0mustbeinvertibleandt)]TJ/F6 7.97 Tf 6.58 0 TD[(1=(1+m0)m0)]TJ/F6 7.97 Tf 8.88 0 TD[(1. Ifn>0fortherightpartoftwists,thetopcolorsare(b;a),andthebottomarea+n0(1)]TJ/F4 11.95 Tf 10.95 0 TD[(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)(a)]TJ/F4 11.95 Tf 10.95 0 TD[(b)anda+(n0+1)(1)]TJ/F4 11.95 Tf 10.95 0 TD[(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)(a)]TJ/F4 11.95 Tf 10.95 0 TD[(b),respectively,anda+(n0+1)(1)]TJ/F4 11.95 Tf -423.96 -20.91 TD[(t)]TJ/F6 7.97 Tf 6.59 0 TD[(1)(a)]TJ/F4 11.95 Tf 12.13 0 TD[(b)=a,wheren=2n0+1.Thelastconditiongives1+n0isinvertibleandt)]TJ/F6 7.97 Tf 6.59 0 TD[(1=n0(1+n0))]TJ/F6 7.97 Tf 6.59 0 TD[(1.Togetherwiththeconditionsofm0,weobtainm0+n0+1063

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Ifnisnegative,similarcalculationsimplythatt)]TJ/F6 7.97 Tf 6.58 0 TD[(1=(1+m0)m0)]TJ/F6 7.97 Tf 8.88 0 TD[(1=(1+n0)n0)]TJ/F6 7.97 Tf 8.88 0 TD[(1andsettingt)]TJ/F6 7.97 Tf 6.59 0 TD[(1=)]TJ/F1 11.95 Tf 9.3 0 TD[(1again,weobtainthesameconclusion.Theothercasesfollowfromsymmetrybythesecases.2 Proof.IfthecolorsoftheNWarcandthearcbetweentheNWandNEarcsarecoloredbya;b2X,thenthecolorsattheSWarcandthearcbetweentheSWarcandtheSEarcarerespectivelya+m(t)(b)]TJ/F4 11.95 Tf 10.46 0 TD[(a)andb+tm(t)(a)]TJ/F4 11.95 Tf 10.45 0 TD[(b)byLemma3.1.1.Thenwemusthavea=a+m(t)(b)]TJ/F4 11.95 Tf 11.24 0 TD[(a)fortheSWarc,andweobtainm(t)(a)]TJ/F4 11.95 Tf 11.23 0 TD[(b)=0.ThenthecolorsoftheNEarcandthearcbetweentheNEandNWarcsare,respectively,a+n(t)(b)]TJ/F4 11.95 Tf 12.18 0 TD[(a)andb+tn(t)(a)]TJ/F4 11.95 Tf 12.18 0 TD[(b)fromtherightntwists.FromthecoloroftheNEarc,wemusthavea+n(t)(b)]TJ/F4 11.95 Tf 11.33 0 TD[(a)=a,sothat(a)]TJ/F4 11.95 Tf 11.33 0 TD[(b)n=0.Thebothequalitiesaresatisedif(a)]TJ/F4 11.95 Tf 11.95 0 TD[(b)h(t)=0inXforh(t)=gcd(m(t);n(t)).264

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Proof.Withthisorientation,thetangleisoftheformoftwocopiesofthemirrorofthetrefoil,andiscolorednon-triviallybythequandleZp[t]=(t2)]TJ/F4 11.95 Tf 11.95 0 TD[(t+1). TheinvariantofthetangleisdeterminedbyProposition4.2.9,buthereweex-hibitamethodtodeterminetheinvariantfromthetablein[41].Forp=2,thetableofquandlecocycleinvariantsin[41]gives16+48utastheinvariantfortre-foilwiththe3-cocycle(x;y;z)=(x)]TJ/F4 11.95 Tf 13.28 0 TD[(y)2(y)]TJ/F4 11.95 Tf 13.28 0 TD[(z).Thisimpliesthatanynon-trivialcoloringcontributesttotheinvariant.Itsmirrorhasthesameproperty.(Notethatthiscasep=2givesthesamevaluesoftheinvariantformirrorimages.)Withtwocopies,anynon-trivialcoloringofthetanglecontributes2t=0whenp=2.Hencetheinvariantvalueofthetangleis64.Fromthetablethisdoesnotembedinknotsupto9crossingsexceptforthefollowingpossibilities:85,810,815;818;819;820;821;916;922;924;925;928;929;930;936;938;939;940;941;942;943;944;945;949. Forp=3,theinvarianttablegives243+486u(2t+2)astheinvariantfortrefoil.Thisimpliesthat486non-trivialcoloringscontributes2t+2totheinvariant.Itsmirrorcontributest+1.Withtwocopies,486non-trivialcoloringsofthetanglecontributes2t+2.Hencetheinvariantvalueofthetangleis243+486u(2t+2).Fromthetablethisdoesnotembedinknotsupto9crossingsexceptfor:31;818;92;94;929;934;938.67

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Forp=7,thetrefoilhas117649astheinvariantvalue,andsodoesthetan-gle.Fromthetablethisdoesnotembedinknotsupto9crossingsexceptfor:31;85;810;811;815;818;819;820;821;91;96;916;923;928;929;938;940. Allcombined,thistangledoesnotembedinknotsupto9crossingsexceptfortheonlypossibilitiesof818;929;938.2Remark4.2.13

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Proof.Thetangle63iswrittenastheadditionR(3)+R()]TJ/F1 11.95 Tf 9.3 0 TD[(3).Henceitiscolorednon-triviallybyZp[t]=(t2)]TJ/F4 11.95 Tf 12.9 0 TD[(t+1)foranyp2Z(weuseonlyprimes),aswellasthedihedralquandleR3.ForthequandleZp[t]=(t2)]TJ/F4 11.95 Tf 12.64 0 TD[(t+1)weusedthe3-cocyclef(x;y;z)=(x)]TJ/F4 11.95 Tf 12.86 0 TD[(y)(y)]TJ/F4 11.95 Tf 12.87 0 TD[(z)p.Thecolorsofthesourceregionforthesetwocopiesofthetrefoildiagrams(R(3)andR()]TJ/F1 11.95 Tf 9.3 0 TD[(3))coincide.Thesignsofthecrossingsareopposite.Hencetheinvariantistrivial,(p2)3copiesof0,forZp[t]=(t2)]TJ/F4 11.95 Tf 13.28 0 TD[(t+1).Computercalculationsareavailableathttp://shell.cas.usf.edu/quandle.Forp=5,inparticular,fromthecalculationsoftheinvariantpostedattheabovewebsite,theTheoremimpliesthatthistanglemayembed,amongknotsinthetableupto9crossings,onlyin:810,812,818,820,924.UsingcalculationswithR3,thetangledoesnotembedin812and818.Thereforethetanglemayembedonlyin810,820,and924. Ontheotherhand,itisseenthat(810)=N(T(63)+R(2;1));(820)=N(T(63)+R(2));(924)=N(T(63)+R(2;2));

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Letxbethecoloroftheboundaryarcs,andlety;x1;x2;x3;x4bethecolorsofthearcsasdepictedinFigure4.13.Wehavex1=t)]TJ/F6 7.97 Tf 6.58 0 TD[(1y+(1)]TJ/F4 11.95 Tf 12.38 0 TD[(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)x=x+t)]TJ/F6 7.97 Tf 6.58 0 TD[(1(y)]TJ/F4 11.95 Tf 12.38 0 TD[(x),x2=ty+(1)]TJ/F4 11.95 Tf 10.55 0 TD[(t)x=x+t(y)]TJ/F4 11.95 Tf 10.56 0 TD[(x),x3=tx+(1)]TJ/F4 11.95 Tf 10.55 0 TD[(t)x1=tx+(1)]TJ/F4 11.95 Tf 10.55 0 TD[(t)x+(t)]TJ/F6 7.97 Tf 6.59 0 TD[(1)]TJ/F1 11.95 Tf 10.55 0 TD[(1)(y)]TJ/F4 11.95 Tf 10.56 0 TD[(x)=x+(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)]TJ/F1 11.95 Tf 12.22 0 TD[(1)(y)]TJ/F4 11.95 Tf 12.22 0 TD[(x),x4=tx+(1)]TJ/F4 11.95 Tf 12.22 0 TD[(t)y=y+t(x)]TJ/F4 11.95 Tf 12.22 0 TD[(y)fromthetopfourcrossings,withthefollowingconditionsthatcomesfromthebottomthreecrossingsxx3=x2;x1x2=x4;xx4=x3:

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andweobtaint+(1)]TJ/F4 11.95 Tf 12.84 0 TD[(t)2=0and(1)]TJ/F4 11.95 Tf 12.84 0 TD[(t)2=(t)]TJ/F6 7.97 Tf 6.58 0 TD[(1)]TJ/F1 11.95 Tf 12.84 0 TD[(1).Bothequationsreducetot2)]TJ/F4 11.95 Tf 13.05 0 TD[(t+1=0.ThereforethetangleiscolorednontriviallybytheAlexanderquandleX=Zp[t;t)]TJ/F6 7.97 Tf 6.59 0 TD[(1]=(t2)]TJ/F4 11.95 Tf 12.29 0 TD[(t+1).Weconcludeourdemonstrationofcalculationsofcoloringsoftanglesbynotingthattheresultsabovearedierentforthesametanglewithdierentorientations.Thuswehadtocarefullyinspecteachpossibleorientations. InasimilarmethodweworkedoutalltheremainingtanglesinthetangletableinFigure4.12.InthetablebelowwelisttheAlexanderquandlesthatcoloredsometanglesinthetangletablenontrivially,andwhichtanglestheycolornon-trivially.72

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Tanglecolored 62,63,717(NWIn,SWIn) 62,63,74(NWIn,NEIn),75(NWIn,NEIn) 76(NWIn,NEIn),77(NWIn,NEIn), 717(NWIn,SWIn) Fortherestofthesection,wegivesomeresultsonothertanglesinthetablethathavebeenobtainedbycocycleinvariants,toshowhowmuchinformationthecocycleinvariantsprovideforothertangles. Tospecifyorientations,weusethenotationalconvention,forexample,\NWIn,SEOut,"toindicatethatthenorthwestarcisorientedinward,andthesoutheastarcisorientedoutward.Tangle76 Thelefthalfofthetangleistrefoil,sothatwetakethequandleZp[t]=(t2)]TJ/F4 11.95 Tf 11.48 0 TD[(t+1).Thenthecolorx2inFig.4.12mustbey.Thentherighthalfofthetanglecanbeclosedtoformthegure-eightknotwiththiscoloring.SincetheAlexanderpolynomialofthegure-eightist2)]TJ/F1 11.95 Tf 11.95 0 TD[(3t+1,wetakep=2tocolorthetanglenon-trivially.73

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(2)ForOrientation(NWIn,NEOut): Forthisorientationthetangle76iscoloredtriviallybyAlexanderquandles,thusweareunabletousethecocycleinvariant.Tangle77 Bythesameargumentasfor76,wehavetousethesamequandle,Zp[t]=(t2)]TJ/F4 11.95 Tf 10.17 0 TD[(t+1).Thusweobtainthesameconclusionsas76. (2)ForOrientation(NWIn,NEOut): Wenoticedthatforthisorientation,thenumeratorofthetangle77istheunknot.Thusthistangleembedsintheunknot,andwearenotabletousecocycleinvariantsincethetangleadmitsonlytrivialcolorings.Tangle713

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(2)ForOrientation(NWIn,NEOut): Thetangleiscolorednon-triviallybyR5.Thetanglehastheinvariantvalue25+50u2+50u3.Thisisprovedasfollows.Theinvariantoftheknot74isasabove.SinceweseethatN(T(713))=74,andthenumberofcoloringsarethesameforthetangleT(713)and74,theinvariantvalueofT(713)mustbethesameasabove.FromthetableforR5,Thistangledoesnotembedinknotsinthetable(upto9crossings)excluding:41;74;816;924;937;939;940;949andtheirmirrors.Thosewedonotknowyetwhetheritembedsare:41;924;937;940,(andpossiblymirrors).Wealsonoticedthat(816)=N(T(713)+R(1)),(939)=N(T(713)+R(1;1)),(949)=N(T(713)+R()]TJ/F1 11.95 Tf 9.3 0 TD[(1;)]TJ/F1 11.95 Tf 9.3 0 TD[(1)).Tangle715 Forthisorientation,wenoticedthat(52)=N(T(715)+R()]TJ/F1 11.95 Tf 9.29 0 TD[(1)),andbycomputingthecolorings,wefoundthatthetangleiscolorednon-triviallybyR7.Thetanglehastheinvariantvalue49+98u3+98u5+98u6.Thisisprovedasfollows.Thetanglecanbenon-triviallycoloredbyR7,andhencesodoes(52)=N(T(715)+R()]TJ/F1 11.95 Tf 9.3 0 TD[(1)).By[41],theinvariantfor52is49+98u3+98u5+98u6.Thenumberofcoloringsis75

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(2)ForOrientation(NWIn,SWOut): Thetangleiscolorednon-triviallybyR7.Thetanglehastheinvariantvalue49+98u+98u2+98u4.Thisisprovedasfollows.Thetanglecanbenon-triviallycoloredbyR7,andhencesodoesitsdenominator(aknotK).SinceKisareducedalternatingdiagram,thecrossingnumberofKis7.By[41],thereareonlytwo7crossingknotsthatarecolorednon-triviallybyR7:71and77.Bothhavethesame(above)invariantvalue.Hencethetanglehasthisinvariantvalue.(TheknotKispresumably77.) FromthetableforR7,Thistangledoesnotembedinknotsinthetable(upto9crossings)excluding:71;77;85;94;912;941. Thetangleiscolorednon-triviallybyR5.Thetanglehastheinvariantvalue25+50u+50u4.Thisisprovedasfollows.Theinvariantoftheknot51isasabove.SinceweseethatD(T(718))=51,andthenumberofcoloringsarethesameforthetangleT(718)and51,theinvariantvalueofT(718)mustbethesameasabove.From76

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(2)ForOrientation(NWIn,SWIn): Thetangleiscolorednon-triviallybyR5.TheinvariantdoesnotdependonthechoiceoforientationforthedihedralquandlewiththeMochizukicocycle[37].Hencetheinvariantisthesameasthecaseaboveand25+50u+50u4.Thenthesameconclusionworksforthiscase,thistangledoesnotembedinknotsinthetable(upto9crossings)excluding:51;818;821;92;912;923;931;940;949,andtheirmirrors.4.2.5EmbeddingDisjointTangles

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381(1+4U+4U2)=27+108U+108U2.Using[41]wecomparethisinvarianttothecocycleinvariantoftheknotsintheknottableupto12crossings,wendthat62t62doesnotembeddinanyknotintheknottable. Morespecically,fromtheinvariantvalue,thenumberofcoloringsof62t62is243,andtheknotinthetableupto12crossingswiththismanynumberof3-coloringsare:12 3339(1+2U)=81+162UwithR3,and62t62doesnotembeddinanyknotontheknottableupto12crossings.(c) 33333=243withR3,and63t63doesnotembeddinanyknotintheknottableupto12crossings.(d)

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Forthecaseoftwistspunknots,thequandlecocycleinvariantsareusedtodetectnon-invertibility.Ourpolynomial3-cocycleprovidedanon-invertibletwist-spuntorusknotthathasnotbeendetectedbydihedralquandles. The2and3-cocylesweconstructedwerealsousedtosolvethetangleembeddingproblems.Thetablesofprimetanglesupto7crossingsandthetableofknotsupto9crossingsarecompared,andthecocycleinvariantsareusedasobstructionstondknotsthatdonotcontainthetanglesinthetable. ThepolynomialcocyclesenableustocomputequandlecocycleinvariantsforalargeclassofAlexanderquandles,andweexpectfurtherdevelopmentsincalculationsandapplications.Inparticular,thesearetheonlyexplicitlyknownhigherdimensionalcocycles,sothatitisexpectedtobeusefulinthestudyofhigherdimensionalquandlecohomologygroupsandapplications.80

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