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Geometry of Link Invariants

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IamindebtedtoProf.A.N.Dranishnikovforhisencouragementandguid-ancethroughmystudyinthegraduateschool,toProf.D.Cenzer,Prof.J.Keesling,Prof.S.ObukhovandProf.Yu.B.Rudyakfortheirinterestinmydissertation,andtothefacultyoftheMathematicsDepartmentfortheirhospitality.Last,butnotleast,IwouldliketothankDr.P.M.AkhmetievforstimulatingcorrespondenceanddiscussionsandJuanLiuforhersupportandpatience. ii

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page ACKNOWLEDGMENTS ................................. ii ABSTRACT ........................................ iv CHAPTER 1INTRODUCTION .................................. 1 2COLOREDFINITETYPEINVARIANTSANDk-QUASI-ISOTOPY ...... 3 3THECONWAYPOLYNOMIAL .......................... 10 4THEMULTI-VARIABLEALEXANDERPOLYNOMIAL ............ 18 5THEHOMFLYANDKAUFFMANPOLYNOMIALS ............... 32 REFERENCES ....................................... 34 BIOGRAPHICALSKETCH ................................ 37 iii

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iv

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v

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RecallthatPLisotopyoflinksinS3canbeviewedastheequivalencerelationgeneratedbyambientisotopyandinsertionoflocalknots.(ThusanyknotinS3isPLisotopictotheunknot.)The\linksmoduloknots"versionoftheVassilievConjectureforknots[ 37 ]couldbeasfollows. 25 ]andTheorem2.2below),whereastheoriginalVassilievConjecturefailsforknotsinanysuchWembeddableinR3(e.g.,intheoriginalWhiteheadmanifold)[ 7 8 ].Similarly,ifMisahomotopysphere,thenM=S3giveneitherapositivesolutionto1.1forknotsinM([ 25 ]andTheorem2.2below),oraproofoftheVassilievConjectureforknotsinM#M[ 7 8 ]. SomefurthergeometrybehindProblem1.1isrevealedby (i)nitetypeinvariants,well-deneduptoPLisotopy,arenotweakerthanniteKL-typeinvariants(cf.x2),well-deneduptoPLisotopy; (ii)indistinguishabilitybyniteKL-typeinvariants,well-deneduptoPLisotopy,impliesk-quasi-isotopyforallk2N; (iii)k-quasi-isotopyforallk2NimpliesTOPisotopyinthesenseofMilnor(i.e.,homotopywithintheclassoftopologicalembeddings); (iv)TOPisotopyimpliesPLisotopy.

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Theassertionfollowssincetheconversetoeachof(i){(iv)holdswithnoproviso:to(ii)byTheorem2.2below,to(iii)byTheorem1.3(b)below,to(i)and(iv)bydenitions. WerecallthatPLlinksL;L0:mS1,!S3(wheremS1=S11ttS1m)arecalledk-quasi-isotopic[ 25 ],iftheyarejoinedbyagenericPLhomotopy,allwhosesingularlinksarek-quasi-embeddings.APLmapf:mS1!S3withpreciselyonedoublepointf(p)=f(q)iscalledak-quasi-embedding,k2N,if,inadditiontothesingletonP0:=ff(p)g,thereexistcompactsubpolyhedraP1;:::;PkS3andarcsJ0;:::;JkmS1suchthatf1(Pj)Jjforeachj6kandPj[f(Jj)Pj+1foreachj
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Weintroducenitetypeinvariantsofcoloredlinks,asastraightforwardgeneralizationoftheKirk{Livingstonsettingofnitetypeinvariantsoflinks[ 16 ].Kirk{Livingstontypekinvariantsarerecoveredastypekinvariantsofcoloredlinkswhosecomponentshavepairwisedistinctcolors,whereastheusualtypekinvariantsoflinks(asdenedbyGoussarov[ 9 ]andStanford[ 33 ])coincidewithtypekinvariantsofmonochromaticlinks.Althoughnormallyeverythingboilsdowntooneofthesetwoextremecases,thegeneralsetting,apartfromitsunifyingappeal,maybeusefulinproofs(cf.proofofProposition3.1)andindealingwithcoloredlinkpolynomials(cf.Theorem4.6).Refertothefollowingbracketedcitationsforabackgroundonnitetypeinvariants:[ 37 10 31 4 33 3 ]. Anm-componentclassicalcoloredlinkmapcorrespondingtoagivencoloringc:f1;:::;mg!f1;:::;mgisacontinuousmapL:mS1!S3suchthatL(S1i)\L(S1j)=;wheneverc(i)6=c(j).Thususuallinkmapscorrespondtotheidentitycoloringandar-bitrarymapstotheconstantcoloring.LetLM31;:::;1(c)bethespaceofallPLc-linkmapsmS1!S3,andletLM(c)denoteitssubspaceconsistingofmapswhoseonlysingularitiesaretransversaldoublepoints(theintegermwillbexedthroughoutthissection).NotethatLM(c)isdisconnectedunlesscisconstant.LetLMn(c)(resp.LM>n(c))denotethesubspaceofLM(c)consistingofmapswithprecisely(resp.atleast)nsingularities.NotethatLM0(c)doesnotdependonc. Givenanyambientisotopyinvariant:LM0(c)!GwithvaluesinanitelygeneratedabeliangroupG,itcanbeextendedtoLM(c)inductivelybytheformula(Ls)=(L+)(L); 3

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withatleastr+1transversaldoublepoints,forsomeniter,theniscalledanitec-typeinvariant,namelyofc-typer.Notethatinournotationanyc-typerinvariantisalsoofc-typer+1.Sometimeswewillidentifyanitec-typeinvariant:LM0(c)!Gwithitsextension:LM(c)!G. Form=1thereisonlyonecoloringc,andnitec-typeinvariants,normalizedby(unknot)=0,coincidewiththeusualGoussarov{Vassilievinvariantsofknots.Ifcisaconstantmap,wewrite\type"insteadof\c-type",andifcistheidentity,wewrite\KL-type"insteadof\c-type",andLMinsteadofLM(c).Ifthecoloringcisacompositionc=de,thenanyd-typekinvariantisalsoac-typekinvariant.Inparticular,anytypekinvariantisaKL-typekinvariant,butnotviceversa.Indeed,thelinkingnumberlk(L)andthegeneralizedSato-Levineinvariant(seex3)areofKL-types0and1,respectively,butoftypesexactly1and3. Itisarguablethattripleandhigher-invariantswithdistinctindicesshouldberegardedashavingKL-type0(astheyassumevaluesindierentcyclicgroupsdependingonlinkhomotopyclassesofpropersublinks,wemusteitheragreetoimaginethesegroupsassubgroups,say,ofZQ=Z,orextendthedenitionofniteKL-typeinvariantstoincludethepossibilitywhereeachpath-connectedcomponentofLMmapstoitsownrange).Incontrast,therearealmostnonitetypeinvariantsoflinkhomotopy[ 24 ],althoughthiscanberemediedbyconsiderationofstringlinks(seereferencesinHughes[ 12 ]orMasbaumandVaintrob[ 23 ])orpartiallydenednitetypeinvariants[ 9 12 ].Weshowinx4thatcertain-invariantsof2-componentlinkshaveintegerliftingsofniteKL-type,atleastsomeofwhicharenotofnitetypedueto Proof. ItisclearthatKL-typerinvariants:LM0!Gformanabeliangroup,whichwedenotebyGr(orG(m)rwhenthenumberofcomponentsneedstobespecied).Clearly,G0isthedirectsumofj0(LM)jcopiesofG.LetGr()denotethesubgroupofGrconsistingofinvariantsvanishingonalllinkswhoselinkhomotopyclassisnot20(LM),and

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onaxedlinkB2withunknottedcomponents.ThelattercanalwaysbeachievedbyaddingaKL-type0invariant,sothequotientmapq:Gr!Gr=G0takesGr()isomorphicallyontoasubgroup,independentofthechoiceofB;moreoverGr=G0=Lq(Gr()).Foranylk2Z,itwasprovedinKirkandLivingston[ 16 ]thatZ(2)1(lk)'Z(generatedbythegeneralizedSato{Levineinvariant)andconjecturedthatZ(2)r(lk)isnotnitelygeneratedforr>1.Itiswell-known[ 33 ]thatthegroupofmonochromaticG-valuedtyperinvariantsofm-componentlinksisnitelygenerated;inparticular,soisG(1)r.Let~GrdenotethesubgroupofGrconsistingofthoseinvariantswhichremainunchangedundertyinglocalknots,i.e.,underPLisotopy.Noticethatthewell-knownproofthattype1invariantsofknotsaretrivialworksalsotoshowthat~G1=G1.InthefollowingremarksweaddressthedierencebetweenGrand~Grforr>1.

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suchthat(L)isunchangedwhenanylocalknotisaddedtoL.Since=(),clearlyBissuchalink. Inparticular,sinceKL-type1invariantsofsingularlinksareinvariantundertyinglocalknotsduetotheone-term(framingindependence)relation,itfollowsthatG2()=~G2()im: Proof. SincetheinclusionP0[f(J0),!P1isnull-homotopic,thereexistsagenericPLhomotopyf0t:f(mS1)!S3suchthatthehomotopyft:=f0tf:mS1!S3satisesf0=f,ft(J0)P1,ft=foutsideJ0,andf1(~J0)isasmallcircle,boundinganembeddeddiskinthecomplementtof1(mS1).Usingtheone-term(framingindependence)relation,weseethatanyniteKL-typeinvariantvanishesonf1.HenceanyKL-type1vanishesonf0=f,whichcompletestheprooffork=1. Beforeproceedingtothegeneralcase,westatethefollowinggeneralizationoftheone-term(framingindependence)relation. Proof.

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thedenitionofk-quasi-isotopy,k>2,wewillnowconstructagenericPLhomotopyf0t1;t:ft1(mS1)!S3suchthatthehomotopyft1;t:=f0t1;tf:mS1!S3satisesft1;0=ft1,ft1;t(J1)P2,ft1;t=ft1outsideJ1,andft1;1takesJ1intoaballB1inthecomplementtoft1;1(mS1nJ1). Indeed,sinceft1(J1)P1[f(J1)isnull-homotopicinP2,wecancontractthe1-dimensionalpolyhedronft1(J1nU),whereUisaregularneighborhoodof@J1inJ1,embeddedbyft1,intoasmallballB01S3nft1(mS1nJ1),byahomotopyft1;tjJ1asrequiredabove.Joiningtheendpointsofft1(mS1n@J1)tothoseofft1;t(J1nU)bytwoembeddedarcsft1;t(U)inanarbitraryway(continuouslydependingont),weobtaintherequiredhomotopy.TakingasmallregularneighborhoodofB01[ft1;1(J1)relativetoft1;1(@J1)weobtaintherequiredballB1. (i)fti1;:::;tin;t=fti1;:::;tinfort=0andoutsideJn; (ii)fti1;:::;tin;t(Jn)Pn+1foreacht; (iii)f1ti1;:::;tin;1(Bn1)=JnforsomePL3-ballBn1. Remark.Infact,theproofofTheorem2.2worksunderaweakerassumptionandwithastrongerconclusion.Namely,k-quasi-isotopycanbereplacedwithvirtualk-quasi-isotopy(seedenitioninMelikhovandRepovs[ 25 ]),whereasindistinguishabilitybyKL-typek 25 ],thehomotopyft1;tcanbevisual-izedasshiftingthearcsf(I0j)ontothearcsF(Ij)andthentakingtheimageofJ1intoaballalongthetrackofthenull-homotopyF.

PAGE 13

invariants(whichcanbethoughtofasaformalgeneralizationofk-equivalenceinthesenseofGoussarov[ 10 ],HabiroandStanford)canbereplacedwithgeometrick-equivalence,denedasfollows.Foreachn>0,letLMn;0denotethesubspaceofLMnconsistingofthelinkmapslsuchthatallsingularitiesoflarecontainedinaballBsuchthatl1(B)isanarc.Wecalllinkmapsl;l02LMngeometricallyk-equivalentiftheyarehomotopicinthespaceLMn[LMn+1;k,whereLMi;kfork>0,i>0consistsofthoselinkmapswithisingularitieswhicharegeometrically(k1)-equivalenttoalinkmapinLMi;0. Toseewhygeometrick-equivalencemaydierfromk-quasi-isotopy,weemploythenotionofweakk-quasi-isotopy,whichisdenedsimilarlytok-quasi-isotopy,butwithnull-homotopiesreplacedbynull-homologies,cf.[ 26 ]. Thedierencebetweengeometrick-equivalenceandweakk-quasi-isotopyisanalogoustothedierencebetweenthelowercentralseriesandthederivedseries,andfork>2canbevisualizedasfollows.LetMkdenotethek-thMilnorlink(cf.,e.g.,Fig.1inMelikhovandRepovs[ 26 ]),andletMWk,k>2,beMkwherethe\long"componentisreplacedbyitsWhiteheaddouble.Unclaspingtheclaspofthedoubledcomponentisa1-quasi-isotopyhtfromMWktotheunlink.Thishtisnota2-quasi-isotopy(evennotaweak2-quasi-isotopy),sinceMkisnotaboundarylink,asdetectedbyCochran'sinvariants(cf.,e.g.,[ 26 ]).ItwouldthusbeinthespiritofConjecture1.1inMelikhovandRepovs[ 26 ]toconjecturethatthereexistsno2-quasi-isotopy,evenweak,fromMWktotheunlink.However,sinceMkis(k1)-quasi-isotopictotheunlink,htrealizesgeometrick-equivalencebetweenMWkandtheunlink. Fork=1nosuchexampleexists,sincegeometric1-equivalenceclearlyimpliesweak1-quasi-isotopy.For2-componentlinkstheconversealsoholds(cf.Fig.2inMilnor[ 27 ]),amplifyingtheanalogywiththelowercentralseriesandthederivedseries,whoserespec-tiveinitialtermscoincide.Thusthedierencebetween1-quasi-isotopyandgeometric1-equivalencefor2-componentlinksreducestothedierencebetween1-quasi-isotopyandweak1-quasi-isotopy(seeFig.2(d)inMelikhovandRepovs[ 25 ]). Existenceofdierencebetweengeometrick-equivalenceand\indistinguishabilitybyKL-typekinvariants,well-deneduptoPLisotopy,"alreadyfork=1,wouldfollow

PAGE 14

fromthe(expected)positivesolutiontoProblem1.5inMelikhovandRepovs[ 25 ]forweak1-quasi-isotopy.

PAGE 15

RecallthattheConwaypolynomialofalinkListheuniquepolynomialrL(z)satisfyingrunknot=1andthecrossingchangeformularL+(z)rL(z)=zrL0(z); whereL+andLdierbyapositivecrossingchange,andL0isobtainedbyorientedsmoothingoftheself-intersectionoftheintermediatesingularlinkLs.NotethatL0hasonemore(respectivelyless)componentthanL+andLiftheintersectionisaself-intersectionofsomecomponent(resp.anintersectionbetweendistinctcomponents). Theskeinrelation(1)showsthatthecoecientatzkinrLisatypekinvariant.Thegeneralizedone-termrelationfornitetypeinvariantsspecializesheretotheequationrL=0foranysplitlinkL(i.e.,alinkwhosecomponentscanbesplitintwononemptypartsbyanembeddedsphere).Using(1)itisnoweasytoseethattheConwaypolyno-mialofanm-componentlinkLisnecessarilyoftheformrL(z)=zm1(c0+c1z2++cnz2n) forsomen.By(1),c0(K)=1foranyknotK,whichcanbeusedtorecursivelyevaluatec0onanylinkL.Forexample,itisimmediatethatc0(L)isthelinkingnumberofLform=2,andc0(L)=ab+bc+caform=3,wherea;b;carethelinkingnumbersofthe2-componentsublinks(cf.[ 22 ]).Forarbitrarym,itiseasytoseethatc0(L)isasymmetricpolynomialinthepairwiselinkingnumbers(andthusaKL-type0invariant);see[ 18 23 ]foranexplicitformula. Proof. 10

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accordingascisontoornot;ineithercase,itisac-type0invariant.Aninductiononiusingtheskeinrelation(1)showsthatthecoecientofrLatzjcj1+iisofc-typei.Inthecasewherecistheidentity,thisisourassertion. InordertocomputerKforaknotK,wecouldconsiderasequenceofcrossingchangesinaplanediagramofK,turningKintotheunknot(cf.[ 31 ]).ThenrK=runknot+zP"irLi,whereLiarethe2-componentlinksobtainedbysmoothingthecrossingsand"i=1arethesignsofthecrossingchanges.WecouldfurtherconsiderasequenceofcrossingchangesinthediagramofeachLi,involvingonlycrossingsofdistinctcomponentsandturningLiintoasplitlink.ThisyieldsrLi=zP"ijrKij,whereKijaretheknotsobtainedbysmoothingthecrossingsand"ij=1arethesignsofthecrossingchanges.SincethediagramofeachKijhasfewercrossingsthanthatofK,wecanexpressrK,iteratingthisprocedure,asPkz2k(Pikirunknot),wherethesignski=1aredeterminedbytheaboveconstruction.Sincerunknot=1,plainlyrK=Pkz2kPiki.NotethatthisprocedureshowsthatrKisindeedapolynomial(ratherthanapowerseries)foranyknotK;asimilarargumentworksforlinks. NowletLbealink,andsupposethatL0isobtainedfromLbytyingaknotKlocallyononeofthecomponents.Wecanechotheaboveconstruction,expressingrL0asPkz2kPikirL,wherethesignskiaresameasabove.Thus 20 ],butweneedtheargumentinordertosetupnotationforuseintheproofofTheorem3.4.

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wherei=ci(i1b1++0bi).HencerL2Z[[z]](ratherthanjustQ[[z1]]),andckk(modgcd(c0;:::;ck1)).Inparticular,form=2weseethat0(L)=c0(L)isthelinkingnumber,and1(L)=c1(L)c0(L)(c1(K1)+c1(K2))is(cf.[ 22 ])thegeneralizedSato{Levineinvariant! UnderthegeneralizedSato{LevineinvariantwemeantheinvariantthatemergedintheworkofPolyak{Viro(see[ 2 ]),Kirk{Livingston[ 16 22 ],Akhmetiev(see[ 1 ])andNakanishi{Ohyama[ 30 ];seealso[ 25 ]. 16 ]).Thisisnotsurprising,becausec1hasKL-type2byProposition3.1,hence(c1)isinvariantunderPL-isotopybyRemark(ii)inx2. (iii)TheextendedrLisinvariantunderTOPisotopyofL.

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assertions(i)and(ii)nowfollowfromTheorem1.3(a),and(iii)eitherfrom[ 25 ]or,alternatively,from(ii)andcompactnessoftheunitinterval. Proposition3.1impliesthateachk(L)isofKL-type2k.Itiseasytocheckthat1(L)=c1(L)c0(L)(Pic1(Ki))is,infact,ofKL-type1andthat2(L)=c2(L)1(L)Xic1(Ki)!c0(L)Xic2(Ki)+Xi6=jc1(Ki)c1(Kj)! Proof. where,asusual,+andstandfortheoverpassandtheunderpass,and0forthesmooth-ingofthecrossings.AssumingthateachLontherighthandsidehasthreecompo-nentsandeachKonlyonecomponent,wecansimplifythisasc0(L000)c0(L++0)c0(L0+0)c0(L00+). Let':S1#S2beagenericC1-approximationwith3doublepointsoftheclockwisedoublecoverS1!S1R2,andletA;B;C;Ddenotethe4boundedcomponentsofR2n'(S1)suchthatDcontainstheorigin.LetKsss2LM13bethecompositionof'andtheinclusionR2S2S3,andletK:S1,!S3nKsss(S1)beaknotinthecomplementofKssslinkingtheclockwiseorientedboundariesofA;B;C;Dwithlinkingnumbersa;b;c;dsuchthata+b+c+2d=0.Finally,deneLsss2LM23tobetheunionofKandKsss,thenlk(Lsss)=a+b+c+2d=0andwendthat2(Lsss)=(a+b+c+d)d(a+c+d)(b+d)(a+b+d)(c+d)(b+c+d)(a+d):

PAGE 19

Thisexpressionisnonzero,e.g.,fora=1,b=2,c=3,d=0. ByProposition3.1andTheorem2.2,n(L)isinvariantunder2n-quasi-isotopy.However,accordingtoProposition3.3,thefollowingstrengtheningofthisassertioncannotbeobtainedbymeansofTheorem2.2. NowletLbeanm-componentlink,andsupposethatL0isobtainedfromLbyagenericPLlinkhomotopyHwithasinglesingularlevelLs.ColorthecomponentsofLwithdistinctcolors,thenthe(m+1)-componentsmoothingL0ofLsisnaturallycoloredwithmcolors.SupposeinductivelythatL0i2:::ik+1isoneofthesmoothedsingularlinks

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inagenericPLcoloredlinkhomotopyH0i2:::ikbetweenthelinkL0i2:::ikandsomelinkL00i2:::ikwhicheitherhasonelesscomponentthanL0i2:::ikorcoincides(geometrically)withL.IfthereexistsuchhomotopiesH0i2:::ilforl=1;:::;2k,wesaythatLsisa+()-skeink-quasi-embedding,providedthattheself-intersectioninHispositive(negative).Askeink-quasi-isotopyisalinkhomotopywhereeverysingularlevelisa+or-skeink-quasi-embedding,dependingonthesignoftheself-intersection. Toseethatnisinvariantunderskeinn-quasi-isotopy Tocompletetheproof,weshowthatn-quasi-isotopyimpliesskeinn-quasi-isotopy.ItsucestoconsideracrossingchangeonacomponentKofL,satisfyingthedenitionofn-quasi-isotopy.Letf,J0;:::;JnandP0;:::;Pnbeasinthedenitionofn-quasi-embedding;wecanassumethatP1containsaregularneighborhoodoff(J0)containingL(J0).WeassociatetoeverylinkL0,suchthatLnKisageometricsublinkofL0,butLitselfisnot,thecollectionofpositiveintegersd(L0)=(d0;:::;dm),wheredifori>0(resp.i=0)istheminimalnumbersuchthattheithcomponentofL0notinLnKisnull-homotopicinPdi(resp.ishomotopictoKwithsupportinPdi).Itiseasytoseethat 2)-quasi-isotopy.

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Supposethatnoneoftheintegersd(L0i2:::ik)exceedsl+1,andatleastrofthemdonotexceedl.Ifr>1,letdibeoneoftheser,thennoneoftheintegers(d(L0i2:::ik)),whereisiorij,exceedsl+1,andatleastr1ofthemdonotexceedl.ThusifatleasttwocomponentsofL0i2:::ikarenotinLnKandnoneoftheintegersd(L0i2:::ik)exceedsl,wecanconstructahomotopyH0i2:::ikasabove,andforeachsingularlinkL0i2:::ik+1inthishomotopyalsoahomotopyH0i2:::ik+1asabove,sothatforeachsingularlinkL0i2:::ik+2inthishomotopy,noneoftheintegersd(L0i2:::ik+2)exceedsl+1.Butitisindeedthecasefork=1,henceforanyoddk,thatatleasttwocomponentsofL0i2:::ikarenotinLnK. Sinceckk(modgcd(c0;:::;ck1)),Theorem3.4impliesthecasel=1,andalsothe2-componentcaseof 26 ]Set=d(l1)(m1) 2e.Theresidueclassofc+kmodulogcdofc;:::;c+k1andall-invariantsoflength6lisinvariantunder(b m1c+k)-quasi-isotopy. (Heredxe=nifx2[n1 2;n+1 2),andbxc=nifx2(n1 2;n+1 2]forn2Z.)

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befoundamongcoecientsofrationalfunctionsinrofthelinkanditstwo-componentsublinks.Itturnsoutthatthisisnotthecasealreadyfork=1.Indeed,wewouldgetanintegerlinkhomotopyinvariantof(monochromatic)nitetype(specically,oftype4),whichisnotafunctionofthepairwiselinkingnumbers(sincec1(123)2(mod1),cf.[ 18 23 ]).Butthisisimpossiblefor3-componentlinks[ 24 ]. Alternatively,onecanarguedirectlyasfollows.Considertheinvariant(L):=1(L)P0(L0)1(L1),wherethesummationisoverallorderedpairs(L0;L1)ofdistinct2-componentsublinksofL.(Recallthatinthe2-componentcase0(L)and1(L)coincidewiththelinkingnumberandthegeneralizedSato-Levineinvariant,respectively.)Itcanbeeasilyveriedthat(L)jumpsby(12)(1;3+)(2;3)+(1;3)(2;3+) 1andr(123)2(mod1),06r<1.

PAGE 23

TheConwaypolynomialisequivalenttothemonochromaticcaseoftheConwaypotentialfunctionLofthecoloredlinkL,namelyrL(xx1)=(xx1)L(x)formonochromaticL.ForalinkLcoloredwithncolors,L2Z[x11;:::;x1n](theringofLaurentpolynomials)ifLhasmorethanonecomponent;otherwiseLbelongstothefractionalideal(xx1)1Z[x1]intheeldoffractionsofZ[x1]. L(x1;:::;xn)isanormalizedversionofthesign-renedAlexanderpolynomialL(t1;:::;tn),whichiswell-deneduptomultiplicationbymonomialsti11:::tinn.IfKisaknot,(xx1)K(x)=xK(x2),whereasforalinkLwithm>1components,L(x1;:::;xn)=x11:::xnnL(x2c(1);:::;x2c(m)),wherec:f1;:::;mgf1;:::;ngisthecoloring,andtheintegers;1;:::;nareuniquelydeterminedbythesymmetryrelationL(x1;:::;xn)=(1)mL(x11;:::;x1n).WerefertoHartley[ 11 ],Traldi[ 35 ]andTuraev[ 36 ]fordenitionofthesign-renedAlexanderpolynomialanddiscussionoftheConwaypotentialfunction. arisesgeometricallyfromconsideringliftsofthegeneratorsofthefundamentalgroupofthelinkcomplementtotheuniversalabeliancover,startingataxedlift~pofthebasepoint.Ifwewishtobasethewholetheoryontheliftsendingat~p,wewillhavetochangetheLeibnizruletoD(fg)=D(f)g1+D(g):

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The\symmetric"LeibnizruleD(fg)=D(f)g1+fD(g)+(f1g1g1f1) doesnotseemtohaveacleargeometricmeaning.(Notetheanalogywiththethreerst-ordernitedierences.)However,itdoescorrespondtoa\symmetric"versionoftheMagnusexpansion,whichwillbebehindthescenesinthissection.Indeed,itsabelianversiongivesrise,inTheorem4.2below,toapowerseriesfL,inthesamewayastheabelianversionoftheusualMagnusexpansionleadstoTraldi'sparametrizationofL,discussedintheproofofTheorem4.8below. 15 ])(a)L(x1;:::;xn)=L(x11;:::;x1n). (b)IfLhasm>1components,thetotaldegreeofeverynonzerotermofLiscongruenttom(mod2). (c)IfLhas>1components,thexi-degreeofeverynonzerotermofLiscongruentmod2tothenumberkiofcomponentsofcoloriplusli:=Plk(K;K0),whereKrunsoverallcomponentsofcoloriandK0overthoseofothercolors. Proof. 35 ]).Thisproves(c),whichimplies(b).Forlinkswith>1components(a)followsfrom(b)andfromthesymmetryrelationinthedenitionofL;forknots{fromtherelationofL(z)withtheConwaypolynomial. ComputationofLismuchharderthanthatofrL,fortheskeinrelation(1)isnolongervalidforarbitrarycrossingchanges.Itsurvives,intheformL+L=(xix1i)L0; inthecasewherebothstrandsinvolvedintheintersectionareofcolori.Thereareotherformulasforpotentialfunctionsoflinksrelatedbylocalmoves,whichsuceforevaluationofonalllinks;seereferencesinMurakami[ 29 ]. Dynnikov[ 6 ]andH.Murakami[ 29 ]noticedthatthecoecientsofthepowerseriesL(eh1=2;:::;ehn=2)are(monochromatic)nitetypeinvariantsofL.

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2t2+:::.TheGaloisgroupofthequadraticequationz=xx1actsonitsrootsbyx7!x1,soLemma4.1(a)impliesthatthecoecientsofthepowerseriesfL(z1;:::;zn):=L(x(z1);:::;x(zn)) areindependentofthechoiceoftherootx. ThesecondassertionfollowsbyMurakami'sproof[ 29 ]oftheresultmentionedabove,andthethirdfromtheskeinrelation(2).Next,sincex1x=(xx1),fL(z1;:::;zn)=L(x(z1)1;:::;x(zn)1)=(1)mL(x(z1);:::;x(zn))=(1)mfL(z1;:::;zn); Thenalassertionmaybenotobvious(since(k+1)-2k1kfork=1;3;7)from(1+4y)1=2=1+4y1Xk=0(1)k

PAGE 26

However,thispowerseriesdoeshaveintegercoecients,sincethisisthecasefor(1+4y)1=2=1+21Xk=1(1)k2k1kyk: Letusconsiderthecaseoftwocolorsinmoredetail.LetcijdenotethecoecientoffL(z1;z2)atzi1zj2.Sincef(z;z)=z1r(z),c00coincideswiththelinkingnumberc0inthe2-componentcaseandiszerootherwise.Theskeinrelation(2)impliesthatfora3-componentlinkK1[K01[K2,coloredasindicatedbysubscript,c10coincideswithlk(K1;K01)(lk(K1;K2)+lk(K01;K2))uptoatype0invariantofthecoloredlink.Thelatterisidenticallyzerobytheconnectedsum whichfollowsfromthedenitionofLinHartley[ 11 ].Sincec01=c0c10orbyananalogousargument,c01=lk(K1;K2)lk(K01;K2).Itfollowsthatfor2-componentlinksc11coincideswiththegeneralizedSato{Levineinvariant1uptoaKL-type0invariant.Since1c11isoftype3,ithastobeadegree3polynomialinthelinkingnumber,whichturnsouttobe1 12(lk3lk).(Sinceunlink=0andHopflink=1,thesecondConwayidentity[ 11 15 ]canbeusedtoevaluateLonaseriesoflinksHnwithlkHn=n.)Thehalf-integerc11isthustheunorientedgeneralizedSato{Levineinvariant,whichisimplicitinthesecondparagraphofKirkandLivingston[ 16 ].Specically,2c11=lk2is(unlesslk=0)theCasson{WalkerinvariantoftheQ-homologysphereobtainedby0-surgeryonthecomponentsofthelink;infact,c11(L)=1(L)+1(L0) 2,whereL0denotesthelinkobtainedfromLbyreversingtheorientationofoneofthecomponents[ 16 ]. Theprecedingparagraphimpliestherstpartofthefollowing

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(i)00=lk,and11istheunorientedgeneralizedSato{Levineinvariant,whichassumesallhalf-integervalues; (ii)1;2k1and2k1;1coincide,whenmultipliedby(1)k+1,withCochran's[ 5 ]derivedinvariantsk,wheneverthelatteraredened(i.e.,lk=0); (iii)ijisoftypei+j;wheni+jisodd,ij=0. 5 ]thatP1i=1iL(z2)i=L(y2),whereLisKojima's-functionandz=yy1.Ontheotherhand,itfollowsfrom[ 13 ]thatL(y2)=(1y2)(1y2)0L(1;y) 13 ](seealsoproofofTheorem4.8(ii)below)thatunderthisassumption0LisaLaurentpolynomial(ratherthanjustarationalfunction).Denef0LbyfL(z1;z2)=z1z2f0L(z1;z2)for2-componentlinkswithlk=0,then1Xi=1(1)iiLz2i=z2f0L(0;z) ItturnsoutthateachcoecientofthepowerseriesfcanbecanonicallysplitintoaQ-linearcombinationofthecoecientsofcertain2n1polynomials.

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(4) 2Z[z1;:::;zn],whereff(x1;:::;xn)gdenotesf(x1;:::;xn)+f(x11;:::;x1n)foranyfunctionf(x1;:::;xn).Moreover,ifn=2l,thecoecientsofP12:::nareinteger. 15 ]:(x;y)=K1(xx1;yy1)+(xy+x1y1)K2(xx1;yy1): whereMisanymonomialinx11;:::;x1n,canbeverieddirectly(separatelyforMofoddandeventotaldegrees),andallowstoexpresseachfxp11:::xpnngintheformX16i1<
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forsomeP0i1:::ik2Z[z1;:::;zn],k>0,andsomeP021 2Z[z1;:::;zn].Thesummandscorrespondingtok=1canbeincludedinP0,andonecangetridofthesummandscorrespondingtooddk>3byrepeateduseoftheformula2fxix1jMg=fxigfx1jMg+fx1jgfxiMg+fxixjgfMg; combinedwith().Notethat()followsimmediatelyfrom()anditsanaloguefxixjMg+fx1ix1jMg=fxixjgfMg: 2P0j1:::j2k. Itremainstoverifyuniquenessof(4).Suppose,bywayofcontradiction,thatanontrivialexpressionQ(x1;:::;xn)intheformoftherighthandsideof(4)isidenticallyzero.ThensoisQ(x1;:::;xn1;xn)Q(x1;:::;xn1;x1n),whichcanberewrittenas[xn]X16i1<
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Repeatingthistwo-stepprocedurebn accordingasnisevenorodd.Consider,forexample,thecaseofoddn.Bysymmetry,P1:::^{:::n=0foreachi.Returningtothepreviousstage[x4]X16i1<
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Moregenerally,wehave Proof.

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LetK1;:::;KmdenotethecomponentsofacoloredlinkL,wherem>1.ItfollowsfromTheorem4.4thatthepowerseriesL(x1;:::;xn):=L(x1;:::;xn) canbeexpressedintheformL(x1;:::;xn)=X16i1<
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compare[ 35 32 ].Thereis,ofcourse,suchasplittingforeveryPi1:::i2n,producingaplethoraofPLisotopyinvariants. LackofananalogueofLemma4.5(ii)forthepowerseriesfLhindersestablishingasimplerelationbetweenthecoecientsoffLandthoseofTraldi'sexpansionofL[ 35 ],whichpreventsonefromexpressingthenitetyperationalsij(discussedinTheorem4.3)asliftingsof-invariants.However,eachijcanbesplitintoalinearcombination(whosecoecientsdependonthelinkingnumber)oftheintegercoecientsofthepowerseries5L,whichdoadmitsuchanexpression. (i)00=lk,and11isanintegerliftingoftheSato{Levineinvariant(1122),butnotanitetypeinvariant; (ii)1;2k1and2k1;1areCochran's[ 5 ]derivedinvariants(1)k+1k,wheneverthelatteraredened(i.e.,lk=0); (iii)ijisanintegerliftingofMilnor's[ 28 ]invariant(1)j(1:::1| {z }i+12:::2| {z }j+1),providedthati+jiseven; (iv)ijisofKL-typei+j; (v)wheni+jisodd,ij=0; (vi)wheni+1,j+1andlkareallevenorallodd,ijiseven; (vii)foragivenL,thereareonlynitelymanypairs(i;j)suchthatij60modulothegreatestcommondivisorijofallkl'swithk6i,l6jandk+l
PAGE 34

whereLdenotestheusualAlexanderpolynomial,`theAlexanderpolynomialofthestringlink[ 19 ],i.e.,theReidemeistertorsionofthebasedchaincomplexofthepair(X;X\R2f0g),whereX=R2Inim`,and`acertainrationalpowerseries,determinedbythe-invariantsof`. forsomeQ1;Q22Z[z1;z2],where=0or1accordingaslk(L)isoddoreven.Setyi=x2i1,thenx2i=1yi+y2iy3i+:::.Theidentitiesfxig2=x2i(x2i1)2=(1yi+y2i:::)y2i;fx1x12gfx1gfx2g=(x21+x22)(x211)(x221)=(2y1y2+y21+y22:::)y1y2 Letusstudythissubstitutionmorecarefully.Letd0ij,d00ijdenotethecoecientsatzi1zj2inPandP12,andletuswrite(k;l)6(i;j)ifk6iandl6j.Thenthecoeciente00ijatyi1yj2inthepowerseriesR2,denedbytheequality(y1y2)1R2(y1;y2)=(z1z2)1P12(z1;z2),isgivenbye00ij=X(k;l)6(i;j)(1)(ik)+(jl)d00kl=(1)i+jX(k;l)6(i;j)d00kl

PAGE 35

Thekeyobservationhereisthatallcoecientsontherighthandsideareevenifi+jiseven. Letusconsiderthecase=1.ThenbyLemma4.5(ii),d00ij=0unlessbothiandjareeven,andd0ij=0unlessbothiandjareodd.Hencee00ij0modulogcdfe00klj(k;l)<(i;j)g,unlessbothiandjareeven,andsimilarlyfore0ij.Nowitfollowsbyinductionthateij2e00ij2d00ijoreij2e0ij2d0ijmoduloEij:=gcdfeklj(k;l)<(i;j)gaccordingasiandjarebothevenorbothodd.Thuseij2(d0ij+d00ij)(modEij)ifi+jiseven.Clearly,thelatterassertionholdsinthecase=0aswell,whichcanbeprovedbythesameargument. Finally,sincelk1(mod2),andx2iareexpressibleaspowerseriesinyiwithintegercoecientsandconstantterm1,thecoecients~eijofTraldi'spowerseries~TL(y1;y2)=(x1x2)lk1LarerelatedtoeijbycongruencemodEij.Thiscompletestheproof,sincebyTraldi[ 35 ],each~eijisanintegerliftingof(1)j+1(1:::1| {z }i+12:::2| {z }j+1). {z }i+12:::2| {z }j+1)=0wheni+jisodd[ 34 ]. 28 ].Henceby(iii),0;2k2k;00(modlk)foreachk.Soiflk=0,everynonzerotermofeitherPorP12involvesbothz1andz2.(Alternatively,thisfollowsfromJin'slemmamentionedintheproofofTheorem4.3(ii).)ByLemma4.5(ii),everynonzerotermofP12hastofurtherincludeeachofthemonceagain,i.e.,P12isdivisiblebyz21z22.Hence,rstly,2k1;1coincideswiththecoecientof2P(z1;z2)atz2k11z2,and,secondly,thiscoecientisnotaectedbyadding(x(z1)x(z2)1+x(z2)x(z1)1)P12(z1;z2)to2P(z1;z2),wherex(z)isasintheproofofTheorem4.2. Proof.

PAGE 36

26 ]Cochran'sinvariantskareinvariantunderk-quasi-isotopy; (b)Milnor'sinvariants(1:::12:::2)ofevenlengthareinvariantunderk-quasi-isotopy,ifeachindexoccursatmost2k+1times. 26 ]All-invariantsoflength62k+3areinvariantunderk-quasi-isotopy.

PAGE 37

WerecallthattheHOMFLY(PT)polynomialandtheDubrovnikversionoftheKaumanpolynomialaretheuniqueLaurentpolynomialsHL;FL2Z[x1;y1]satisfyingHunknot=Funknot=1andxHL+x1HL=yHL0;xFL+x1FL=y(FL0xw(L1)w(L0)FL1); 20 ]areobtainedasHL(ia;iz)and(1)m1FL(ia;iz). (i)ForeachL:mS1,!S3andanyn2Nthereexistsan"n>0suchthatifL0:mS1,!S3isC0"n-closetoL,HL0(ech=2;eh=2eh=2)HL(ech=2;eh=2eh=2)mod(hn);FL0(e(c1)h=2;eh=2eh=2)FL(e(c1)h=2;eh=2eh=2)mod(hn): (iii)TheextendedHLandFLareinvariantunderTOPisotopyofL.

PAGE 38

20 ]implythatHandFareinvariantunderPLisotopy.(NotethattheconnectedsuminLickorish[ 20 ]isHashizume's,notthecomponentwiseconnectedsumofMelikhovandRepovs[ 25 ].)Ontheotherhand,itwasnoticedinLieberum[ 21 ](compare[ 9 4 33 ])thatthecoecientsofthepowerseriesHL(ech=2;eh=2eh=2)=1Xk=0k+m1Xi=0pkicihk2Q[c][[h]]FL(e(c1)h=2;eh=2eh=2)=1Xk=0k+m1Xi=0qkicihk2Q[c][[h]] are(monochromatic)nitetypeinvariantsofL.Specically,eachpkiandeachqkiisoftypek,moreoverp0i=q0i=m1;i(theKroneckerdelta).(TheargumentinLieberum[ 21 ]wasforHLHT2andFLFT2,whereT2denotesthetrivial2-componentlink,butitworksaswellforHLandFL,compare[ 3 ].)TherestoftheproofrepeatsthatofTheorem3.2.

PAGE 39

[1] P.M.Akhmetiev,D.Repovs,AgeneralizationoftheSato{Levineinvariant,in:TrudyMat.Inst.im.Steklova221(1998),69{80;Englishtransl.,in:Proc.SteklovInst.Math.221(1998),60{70. [2] P.M.Akhmetiev,J.Malesic,D.Repovs,AformulaforthegeneralizedSato{Levineinvariant,Mat.Sbornik192no.1(2001),3{12;Englishtransl.,Sb.Math.(2001),1{10. [3] D.Bar-Natan,OntheVassilievknotinvariants,Topology34(1995),423{472. [4] J.S.Birman,X.-S.Lin,KnotpolynomialsandVassiliev'sinvariants,Invent.Math.111(1993),225{270. [5] T.Cochran,Geometricinvariantsoflinkcobordism,Comm.Math.Helv.60(1985),291{311. [6] I.A.Dynnikov,TheAlexanderpolynomialinseveralvariablescanbeexpressedintermsofVassilievinvariants,UspekhiMat.Nauk52(1997)no.1,227{228;Englishtransl.,Russ.Math.Surv.52(1997)no.1,219{221. [7] M.Eisermann,InvariantsdeVassilievetConjecturedePoincare,ComptesRendusAcad.Sci.Paris,Ser.I334(2002),1005{1010. [8] M.Eisermann,VassilievinvariantsandthePoincareConjecture,Topology43(2004),1211-1229. [9] M.N.Goussarov,AnewformoftheConway{Jonespolynomialoforientedlinks,Zap.Nauch.Sem.LOMI193\Geom.iTopol.1"(1991),4{9;Englishtransl.,in:TopologyofManifoldsandVarieties,Adv.SovietMath.18,Amer.Math.Soc.,ProvidenceRI(1994),167{172. [10] M.N.Goussarov,Variationsofknottedgraphs.Geometrictechniquesofn-equivalence,AlgebraiAnaliz12no.4(2000),79{125;Englishtransl.,St.-PetersburgMath.J.12(2000),569{604. [11] R.Hartley,TheConwaypotentialfunctionforlinks,Comm.Math.Helv.58(1983),365{378. [12] J.R.Hughes,Finitetypelinkhomotopyinvariantsofk-triviallinks,J.KnotTheoryRam.12no.3(2003),375{393. [13] J.T.Jin,OnKojima's-function,in:DierentialTopology(Proceedings,Siegen1987)(U.Koschorke,ed.),Lect.NotesMath.,Springer-Verlag,Berlin1350,14{30(1988). 34

PAGE 40

[14] T.Kadokomi,Seifertcomplexforlinksand2-variableAlexandermatrices,in:Knots96(S.Suzuki,ed.),WorldSci.Publ.Co.,Singapore(1997),395{409. [15] M.E.Kidwell,Onthetwo-variableConwaypotentialfunction,Proc.Amer.Math.Soc.98(1986),485{494. [16] P.Kirk,C.Livingston,VassilievinvariantsoftwocomponentlinksandtheCasson-Walkerinvariant,Topology36no.6(1997),1333{1353. [17] P.Kirk,C.Livingston,Z.Wang,TheGassnerrepresentationforstringlinks,Comm.Cont.Math.3(2001),87{136. [18] J.Levine,TheConwaypolynomialofanalgebraicallysplitlink,in:Knots'96(S.Suzuki,ed.),WorldSci.Publ.Co.,Singapore(1997),23{29. [19] J.-Y.LeDimet,Enlacementsd'intervallesettorsiondeWhitehead,Bull.Soc.Math.France129(2001),215{235. [20] W.B.R.Lickorish,Anintroductiontoknottheory,Grad.TextsinMath.175,Springer,NewYork(1997). [21] J.Lieberum,ThenumberofindependentVassilievinvariantsintheHOMFLYandKaumanpolynomials,DocumentaMath.8(2003),275{298. [22] C.Livingston,Enhancedlinkingnumbers,Amer.Math.Monthly110(2003),361{385. [23] G.Masbaum,A.Vaintrob,Milnornumbers,spanningtrees,andtheAlexander{Conwaypolynomial,Adv.Math.180(2003),765{797. [24] B.Mellor,D.Thurston,Ontheexistenceofnitetypelinkhomotopyinvariants,J.KnotTheoryRam.10(2001),1025{1040. [25] S.A.Melikhov,D.Repovs,k-quasi-isotopy:I.Questionsofnilpotence,J.KnotTheoryRam.(inpress). [26] S.A.Melikhov,D.Repovs,k-quasi-isotopy:II.Comparison,J.KnotTheoryRam.(inpress). [27] J.Milnor,Linkgroups,Ann.ofMath.59(1954),177{195. [28] J.Milnor,Isotopyoflinks,in:AlgebraicGeometryandTopology:ASymposiuminHonorofS.Lefschetz(R.H.Fox,D.Spencer,J.W.Tucker,eds.),PrincetonUniv.Press,Princeton(1957),208{306. [29] H.Murakami,Aweightsystemderivedfromthemulti-variableConwaypotentialfunction,J.LondonMath.Soc.59(1999),698{714. [30] Y.Nakanishi,Y.Ohyama,Deltalinkhomotopyfortwocomponentlinks,III,J.Math.Soc.Japan55no.3(2003),641{654. [31] V.V.Prasolov,A.B.Sossinsky,Knots,links,braidsand3-manifolds,MCCME,Moscow(1997);Englishtransl.,Trans.Math.Monogr.,Amer.Math.Soc.,ProvidenceRI154(1997).

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[32] D.Rolfsen,LocalizedAlexanderinvariantsandisotopyoflinks,Ann.ofMath.(2)101no.1(1975),1{19. [33] T.Stanford,Finitetypeinvariantsofknots,linksandgraphs,Topology35(1996),1027{1050. [34] L.Traldi,Milnor'sinvariantsandthecompletionsoflinkmodules,Trans.Amer.Math.Soc.284(1984),401{424. [35] L.Traldi,Conway'spotentialfunctionanditsTaylorseries,KobeJ.Math.5(1988),233{264. [36] V.Turaev,Introductiontocombinatorialtorsions,Birkhauser,Basel(2001). [37] V.A.Vassiliev,Complementsofdiscriminantsofsmoothmaps:TopologyandApplications,Trans.Math.Monogr.,Amer.Math.Soc.,ProvidenceRI98(1992).

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IwasborninMoscow,Russia,in1980.MyhighschoolwastheMoscowStateFifty-SeventhSchool,whereIstudiedinamathematicallyorientedclassin1992{96.In1996{2001IcontinuedmyeducationattheMechanicsandMathematicsDepart-mentoftheMoscowStateUniversity,whichconferredabachelor'sdegreeinpureandappliedmathematicsonme.IhavebeenintheGraduateSchooloftheUniversityofFloridasince2001. 37


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GEOMETRY OF LINK INVARIANTS


By

SERGEY A. MELIKHOV



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005















ACKNOWLEDGMENTS

I am indebted to Prof. A. N. Dranishnikov for his encouragement and guid-

ance through my study in the graduate school, to Prof. D. Cenzer, Prof. J. Keesling,

Prof. S. Obukhov and Prof. Yu. B. Rudyak for their interest in my dissertation, and to the

faculty of the Mathematics Department for their hospitality.

Last, but not least, I would like to thank Dr. P. M. Akhmetiev for stimulating

correspondence and discussions and Juan Liu for her support and patience.















TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. ii

ABSTRACT . . . . . . . . . . iv

CHAPTER

1 INTRODUCTION .................................. 1

2 COLORED FINITE TYPE INVARIANTS AND k-QUASI-ISOTOPY ...... 3

3 THE CONWAY POLYNOMIAL .......... ................ 10

4 THE MULTI-VARIABLE ALEXANDER POLYNOMIAL ........... .18

5 THE HOMFLY AND KAUFFMAN POLYNOMIALS ............. 32

REFERENCES ................................. .... 34

BIOGRAPHICAL SKETCH ..................... .......... 37















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

GEOMETRY OF LINK INVARIANTS

By

Sergey A. Melikhov

August 2005

C'!I ii: Alexander N. Dranishnikov
A, i ,r Department: Mathematics

k-Quasi-isotopy is an equivalence relation on piecewise-linear (or smooth) links in the

Euclidean 3-space, whose definition closely resembles the first k stages of the construction

of a Casson handle. It generalizes and refines both k-cobordism of Cochran and Orr and

k-linking of S. Eilenberg, as modified by N. Smythe and K. Kc.1 li- i-li k-Quasi-isotopy

turned out to be helpful for understanding the geometry of the Alexander polynomial and

Milnor's p-invariants, leading, in particular, to the following results.

There exists a direct multi-variable analogue of the Conway polynomial VL(z),

i.e., a (finite) polynomial VL(z,,... Zn) with integer coefficients, satisfying L -

VL = ZiVLo for any Conway triple (L+, L_, Lo) whose disagreement is within the

components of the i-th color. Moreover, for 2-component links all coefficients of the

2-variable VL are integer lifting of Milnor's invariants p(1 ... 12... 2) of even length.

Each Cochran's derived invariant /3 (defined originally when the linking number

vanishes) extends to a Vassiliev invariant of 2-component links.

If Ki are the components of a link L, each coefficient of the power series VL/ H V7K

is invariant under sufficiently close Co-approximation, and can be (uniquely)

extended, preserving this property, to all topological links in R3. The same holds for

the power series VLI(z,..., z,)/ H VKj(z), as well as HL/ HK, and FL/H FKi,

where HL and FL are certain exponential parameterizations of the two-variable

HOMFLY and Kauffman polynomials.









* No difference between PL isotopy and TOP isotopy (as equivalence relations on PL

links in R3) can be detected by Vassiliev invariants.














CHAPTER 1
INTRODUCTION

Recall that PL isote'ip; of links in S3 can be viewed as the equivalence relation

generated by ambient isotopy and insertion of local knots. (Thus any knot in S3 is PL

isotopic to the unknot.) The "links modulo knots" version of the Vassiliev Conjecture for

knots [37] could be as follows.

Problem 1.1. Are PL isote'i~; classes of links in S3 separated by finite type invariants

that are well-,,. ,,. .1 up to PL isote'i,?

It is to be noted that both questions depend heavily on the topology of S3 (or R3)

as opposed to other 3-manifolds. Indeed, the answer to 1.1 is negative for links (in fact,

even knots) in any contractible open 3-manifold W other than R3 ([25] and Theorem

2.2 below), whereas the original Vassiliev Conjecture fails for knots in any such W

embeddable in R3 (e.g., in the original Whitehead manifold) [7, 8]. Similarly, if M is a

homotopy sphere, then M S3 given either a positive solution to 1.1 for knots in M ([25]

and Theorem 2.2 below), or a proof of the Vassiliev Conjecture for knots in M#M [7, 8].

Some further geometry behind Problem 1.1 is revealed by

Theorem 1.2. The answer to 1.1 is affirmative if and only if the following four state-

ments (on PL links in S3) hold simultii, ..u-l;,

(i) finite type invariants, well-I, ., ,..1 up to PL isote'i,; are not weaker than finite
KL-type invariants (cf. 2), well-,. .,, .1 up to PL isote'i~

(ii) i .i.,- /li,i,-.i]il -/lii by finite KL-'i:,,,. invariants, well-,., i,, ,l up to PL isotei,',

implies k-quasi-isotei'; for all k E N;

(iii) k-quasi-isotclji, for all k E N implies TOP '-:,'/,,;;, in the sense of Milnor (i.e.,

homote'i.; within the class of to'', 1 I. ':. l embeddings);

(iv) TOP isot,'pi, implies PL '-. 4.'./,/









The assertion follows since the converse to each of (i)-(iv) holds with no proviso: to

(ii) by Theorem 2.2 below, to (iii) by Theorem 1.3(b) below, to (i) and (iv) by definitions.

We recall that PL links L, L': mS1 S3 (where mS1 = S' U ... U S'J are called

k-quasi-isotopic [25], if they are joined by a generic PL homotopy, all whose singular

links are k-quasi-embeddings. A PL map f: mS1 -> S3 with precisely one double

point f(p) = f(q) is called a k-quasi-embedding, k E N, if, in addition to the singleton

Po := {f(p)}, there exist compact subpolyhedra Pi,..., Pk C S3 and arcs Jo,... Jk C mS'

such that f-'(Pj) C Jj for each j k and Pj U f(Jj) C Pj+i for each j < k, where the

latter inclusion is null-homotopic for each j < k.

This definition is discussed in detail and illustrated by examples in Melikhov and

Repos [25] and [26]. The only result we need from these papers is the following obvious

Theorem 1.3. [25] (a) For each k E N, k-quasi-isote'i', classes of all suff.- ill, close PL

approximations to a t('l'. 1..':. 'l link coincide.

(b) TOP isotopic PL links are k-quasi-isotopic for all k E N.

The paper is organized as follows. 2 contains a little bit of general geometric theory

of finite KL-type invariants, which we interpret in the context of colored links. (We do

not attempt at extending any of the standard algebraic results of the theory of finite type

invariants.) The rest of the paper is devoted to specific examples. Except for invariants

of genuine finite type -from the Conway polynomial (3), the multi-variable Alexander

polynomial (Theorems 4.2 and 4.3) and the HOMFLY and Kauffman polynomials (5),

we need to work just to produce interesting examples (Theorems 4.4 and 4.6). The

p ,-off is a further clarification of the relationship between the Alexander polynomial and

geometrically more transparent invariants of Milnor [28] and Cochran [5] (Theorem 4.8).

We are also able to extract additional geometric information (Theorem 3.4 and Corollary

4.10b), which was the initial motivation for this paper (compare Problem 1.5 in Melikhov

and Repovs [25]).














CHAPTER 2
COLORED FINITE TYPE INVARIANTS AND k-QUASI-ISOTOPY

We introduce finite type invariants of colored links, as a straightforward generalization

of the Kirk-Livingston setting of finite type invariants of links [16]. Kirk-Livingston

type k invariants are recovered as type k invariants of colored links whose components

have pairwise distinct colors, whereas the usual type k invariants of links (as defined by

Goussarov [9] and Stanford [33]) coincide with type k invariants of monochromatic links.

Although normally everything boils down to one of these two extreme cases, the general

setting, apart from its unifying appeal, may be useful in proofs (cf. proof of Proposition

3.1) and in dealing with colored link polynomials (cf. Theorem 4.6). Refer to the following

bracketed citations for a background on finite type invariants: [37, 10, 31, 4, 33, 3].

An m-component classical colored link map corresponding to a given coloring

c: {1,...,m} {1,...,m} is a continuous map L: mS1 -- S3 such that L(S/')Ln (S) = 0

whenever c(i) / c(j). Thus usual link maps correspond to the identity coloring and ar-

bitrary maps to the constant coloring. Let LM,3...,1(c) be the space of all PL c-link maps

mS1 -- S3, and let M(c) denote its subspace consisting of maps whose only singularities

are transversal double points (the integer m will be fixed throughout this section). Note

that M(c) is disconnected unless c is constant. Let LIM (c) (resp. LM>,(c)) denote the

subspace of LM(c) consisting of maps with precisely (resp. at least) n singularities. Note

that L./i,(c) does not depend on c.

Given any ambient isotopy invariant X : C I,,(c) -- G with values in a finitely

generated abelian group G, it can be extended to LM(c) inductively by the formula


X(L,) X(L+)- X(L),

where L+, L_ e L (c) differ by a single positive crossing of components of the same

color, and L E C 1 ,I (c) denotes the intermediate singular link with the additional

double point. If thus obtained extension X vanishes on LM>r+i(c), i.e., on all c-link maps









with at least r + 1 transversal double points, for some finite r, then X is called a finite

c-type invariant, namely of c-type r. Note that in our notation any c-type r invariant is

also of c-type r + 1. Sometimes we will identify a finite c-type invariant X: L/, (c) -- G

with its extension X: LM(c) -- G.

For m 1 there is only one coloring c, and finite c-type invariants, normalized by

x(unknot) = 0, coincide with the usual Goussarov-Vassiliev invariants of knots. If c is a

constant map, we write "type" instead of "c-type", and if c is the identity, we write "KL-

type" instead of "c-type", and LM, instead of LM (c). If the coloring c is a composition

c = de, then any d-type k invariant is also a c-type k invariant. In particular, any type k

invariant is a KL-type k invariant, but not vice versa. Indeed, the linking number lk(L)

and the generalized Sato-Levine invariant (see 3) are of KL-types 0 and 1, respectively,

but of types exactly 1 and 3.

It is arguable that triple and higher p-invariants with distinct indices should be

regarded as having KL-type 0 (as they assume values in different cyclic groups depending

on link homotopy classes of proper sublinks, we must either agree to imagine these groups

as subgroups, -iv, of Z E Q/Z, or extend the definition of finite KL-type invariants to

include the possibility where each path-connected component of LM maps to its own

range). In contrast, there are almost no finite type invariants of link homotopy [24],

although this can be remedied by consideration of string links (see references in Hughes

[12] or Masbaum and Vaintrob [23]) or partially defined finite type invariants [9, 12]. We

show in 4 that certain p-invariants of 2-component links have integer lifting of finite

KL-type, at least some of which are not of finite type due to

Proposition 2.1. The KL-type 0 invariant v(L) = (-l)lk(L) is not offinite type.


Proof. By induction, v(Ls) = 2k for any singular link L8 with k transversal intersections

between distinct components and no self-intersections. O

It is clear that KL-type r invariants X : C -- G form an abelian group, which we

denote by G, (or G' ) when the number of components needs to be specified). Clearly, Go

is the direct sum of 7Io(CM)l copies of G. Let G,(A) denote the subgroup of Gr consisting

of invariants vanishing on all links whose link homotopy class is not AE 7 o(CM), and









on a fixed link B\ E A with unknotted components. The latter can alv--, be achieved

by adding a KL-type 0 invariant, so the quotient map q: Gr -- G,/Go takes G,(A)

isomorphically onto a subgroup, independent of the choice of B%; moreover G,/Go

%x q(G,(A)). For any 1k E Z, it was proved in Kirk and Livingston [16] that Z 2)(lk) Z
(generated by the generalized Sato-Levine invariant) and conjectured that Z 2)(1k) is

not finitely generated for r > 1. It is well-known [33] that the group of monochromatic

G-valued type r invariants of m-component links is finitely generated; in particular, so

is Gr Let Gr denote the subgroup of Gr consisting of those invariants which remain

unchanged under tying local knots, i.e., under PL isotopy. Notice that the well-known

proof that type 1 invariants of knots are trivial works also to show that G1 = G1. In the

following remarks we address the difference between Gr and Gr for r > 1.

Remarks. (i). By evaluating any type r invariant of knots on the components of a link

L E A we obtain a monomorphism

A: eC)G GCr(A).
i= 1

We claim that its image (whose elements we regard here as the "least interesting among

all) is a direct summand, with complement containing Gr(A). Indeed, for any X E Gr(A)

define a knot invariant Xi by i(K) x= (Ki), where K, is the link, obtained by tying

the knot K locally on the ith component of B\. Then Xi is a type r knot invariant, since

the local knot on the ith component of Ki, viewed as a knotted ball pair (B3, B1), can

be chosen to look precisely the same as K outside a small ball. Define an endomorphism

4) of Gr(A) by Q(X)(L) X(L) EC~ L (i ), where Lt denotes the ith component of

L. Then 4) takes any X E imA to zero and any X E Gr(A) to itself, and consequently

defines a splitting (depending, in general, on the choice of B\) of the quotient map

G,(A) cokerA, with image containing G,(A).

(ii). We claim that an invariant X in the complement to imA is invariant under tying local

knots iff the restriction of X to 1M1 (which is a KL-type r 1 invariant of singular links)

is. Indeed, the "only if" implication is trivial. Conversely, it suffices to find one link L E A








such that X(L) is unchanged when any local knot is added to L. Since X = t(X), clearly

B\ is such a link.
In particular, since KL-type 1 invariants of singular links are invariant under tying

local knots due to the one-term (framing independence) relation, it follows that

G2(A) G2(A) imA.




Theorem 2.2. Let X be a KL-type k invariant. If x is invariant under PL isotei,;, then

it is invariant under k-quasi-isot(i',1

f. For k = 0 there is nothing to prove, so assume k > 1. It suffices to show that any

x E Gk vanishes on any k-quasi-embedding f: mS1 S3 with precisely one double point

f(p) f(q). Let Po,..., Pk and Jo,..., Jk-1 be as in the definition of k-quasi-embedding,
and let Jo denote the subarc of Jo with 9Jo = {p, q}. In order to have enough room

for general position we assume that Pi's are compact 3-manifolds (with boundary) and

Jo C Int Jo (this can be achieved by taking small regular neighborhoods).
Since the inclusion Po U f(Jo) c- P1 is null-homotopic, there exists a generic PL

homotopy fJ: f(mS1) -S S3 such that the homotopy ft:= f"f: mS1 S3 satisfies fo =f,

ft(Jo) C Pi, ft = f outside Jo, and f1(Jo) is a small circle, bounding an embedded disk in
the complement to fi(mS1). Using the one-term (framing independence) relation, we see

that any finite KL-type invariant vanishes on fl. Hence any KL-type 1 vanishes on fo = f,
which completes the proof for k = 1. O

Before proceeding to the general case, we state the following generalization of the

one-term (framing independence) relation.

Lemma 2.3. If f C 1 has all its double points inside a ball B such that f- (B) is an

arc, then for ,1:,1 x E G,, x(f) = 0.

F *' By induction on n. O

Proof of 2.2 continuedd). Let us study the jump of our invariant X E Gk on the link maps

ft occurring at singular moments of the homotopy f[. Let ft, E C 11 be one. Using








the definition of k-quasi-isotopy, k > 2, we will now construct a generic PL homotopy

ftl,t: ft,(mS1) S3 such that the homotopy ft,,t := f',tf mS S3 satisfies ft,o = fti,
fti,t(J1) C P2, fti,t = ft outside Ji, and fti,i takes J1 into a ball B1 in the complement to
ft,,i(nS1 \ Ji).
Indeed, since ft (Ji) C P1 U f(J1) is null-homotopic in P2, we can contract the
1-dimensional polyhedron ft, (J1 \ U), where U is a regular neighborhood of &JI in Ji,
embedded by ft,, into a small ball B' C S3 \ ft,(mS1 \ Ji), by a homotopy ft,,tl j as
required above. Joining the endpoints of ft,(mS1 \ J1i) to those of fti,t(Ji \ U) by two
embedded arcs ft,,t(U) in an arbitrary way (continuously depending on t), we obtain the
required homotopy. Taking a small regular neighborhood of B, U fti,,(J1) relative to

ft1,l(aJi) we obtain the required ball B1.1
By Lemma 2.3, any finite KL-type invariant vanishes on ft,il. The homotopy ft,,t as
well as the analogously constructed ft/,t for each ft, c C11 does not change any invariant
of KL-type 2, so for k = 2 we are done. The proof in the general case must be transparent
now. For completeness, we state E

2.4. The nth inductive step. For each critical level ft, ....,, E C V i of I,,:. of
the homotopies constructed in the previous step, there exists a generic PL homote'l':

fti,(....t,t -i ft;s .... t (nS1) S3 such that the homoter'i fti .....tj,t .= fl^....t,,,ttft,....ti,


(i) ft ,....,tt = ft, ....t, for t = 0 and outside J,;
(ii) ftj,... t,t(j,) C Pn+1 for each t;
(ii) f ...t (B -) J= for some PL 3-ball Bn-1. O

Remark. In fact, the proof of Theorem 2.2 works under a weaker assumption and with a
stronger conclusion. Namely, k-quasi-isotopy can be replaced with virtual k-quasi-isotopy

(see definition in Melikhov and Repovs [25]), whereas indistinguishability by KL-type k



1 Using the notation from Melikhov and Repovs [25], the homotopy ft,,t can be visual-
ized as shifting the arcs f(I) onto the arcs F(Ij) and then taking the image of J1 into a
ball along the track of the null-homotopy F.









invariants (which can be thought of as a formal generalization of k-equivalence in the sense

of Goussarov [10], Habiro and Stanford) can be replaced with geometric k-equivalence,

defined as follows. For each n > 0, let L.1 ,, denote the subspace of C11 consisting of

the link maps I such that all singularities of I are contained in a ball B such that 1 (B) is

an arc. We call link maps 1,1' E C11 geome l,.:. a'll k-equivalent if they are homotopic in

the space C 1 U C 1, i I;k, where C 11, for k > 0, i > 0 consists of those link maps with i

singularities which are geometrically (k 1)-equivalent to a link map in C 1 ,,

To see why geometric k-equivalence may differ from k-quasi-isotopy, we employ the

notion of weak k-quasi-isotopy, which is defined similarly to k-quasi-isotopy, but with

null-homotopies replaced by null-homologies, cf. [26].

The difference between geometric k-equivalence and weak k-quasi-isotopy is analogous

to the difference between the lower central series and the derived series, and for k > 2

can be visualized as follows. Let i 1, denote the k-th Milnor link (cf., e.g., Fig. 1 in

Melikhov and Repovs [26]), and let M7A k > 2, be 11, where the "1.,i,; component

is replaced by its Whitehead double. Unclasping the clasp of the doubled component

is a 1-quasi-isotopy ht from 1W to the unlink. This ht is not a 2-quasi-isotopy (even

not a weak 2-quasi-isotopy), since i i, is not a boundary link, as detected by Cochran's

invariants (cf., e.g., [26]). It would thus be in the spirit of Conjecture 1.1 in Melikhov and

Repovs [26] to conjecture that there exists no 2-quasi-isotopy, even weak, from M3 to the

unlink. However, since il, is (k 1)-quasi-isotopic to the unlink, ht realizes geometric

k-equivalence between MW and the unlink.

For k = 1 no such example exists, since geometric 1-equivalence clearly implies weak

1-quasi-isotopy. For 2-component links the converse also holds (cf. Fig. 2 in Milnor [27]),

amplifying the analogy with the lower central series and the derived series, whose respec-

tive initial terms coincide. Thus the difference between 1-quasi-isotopy and geometric

1-equivalence for 2-component links reduces to the difference between 1-quasi-isotopy and

weak 1-quasi-isotopy (see Fig. 2(d) in Melikhov and Repovs [25]).

Existence of difference between geometric k-equivalence and "indistinguishability by

KL-type k invariants, well-defined up to PL isotopy," already for k = 1, would follow







9

from the (expected) positive solution to Problem 1.5 in Melikhov and Repovs [25] for weak

1-quasi-isotopy.














CHAPTER 3
THE CONWAY POLYNOMIAL

Recall that the Co,', '1.;/ / -;;,,I I.:Il/ of a link L is the unique polynomial VL(z)

satisfying unknot = 1 and the crossing change formula


VL () vL (z) = ZV (), (1)

where L+ and L_ differ by a positive crossing change, and Lo is obtained by oriented

smoothing of the self-intersection of the intermediate singular link L,. Note that Lo

has one more (respectively less) component than L+ and L_ if the intersection is a

self-intersection of some component (resp. an intersection between distinct components).

The skein relation (1) shows that the coefficient at zk in VL is a type k invariant. The

generalized one-term relation for finite type invariants specializes here to the equation

L = 0 for any split link L (i.e., a link whose components can be split in two nonempty

parts by an embedded sphere). Using (1) it is now easy to see that the Conway polyno-

mial of an m-component link L is necessarily of the form


VL(z) z= zm-(co + C1z2 +- C+z2n)

for some n. By (1), co(K) = 1 for any knot K, which can be used to recursively evaluate

co on any link L. For example, it is immediate that co(L) is the linking number of L for

m = 2, and co(L) = ab + be + ca for m = 3, where a, b, c are the linking numbers of the 2-

component sublinks (cf. [22]). For arbitrary m, it is easy to see that co(L) is a symmetric

polynomial in the pairwise linking numbers (and thus a KL-type 0 invariant); see [18, 23]

for an explicit formula.

Proposition 3.1. ck(L) is of KL-;1,1,' 2k.


Proof. Let L be a c-colored link, and let Icl denote the number of colors used, i.e., the

cardinality of the image of c. Then the coefficient of VL at z1 l-1 is either co(L) or 0









according as c is onto or not; in either case, it is a c-type 0 invariant. An induction on i

using the skein relation (1) shows that the coefficient of 7L at z 1l-1+i is of c-type i. In the

case where c is the identity, this is our assertion. E

In order to compute VK for a knot K, we could consider a sequence of crossing

changes in a plane diagram of K, turning K into the unknot (cf. [31]). Then VK

Vunknot + z L CiL,, where Li are the 2-component links obtained by smoothing the
crossings and Ei = 1 are the signs of the crossing changes. We could further consider a

sequence of crossing changes in the diagram of each Li, involving only crossings of distinct

components and turning Li into a split link. This yields L, = z xCEij~Ky, where Kij are

the knots obtained by smoothing the crossings and ij = 1 are the signs of the crossing

changes. Since the diagram of each Kij has fewer crossings than that of K, we can express

VK, iterating this procedure, as Yk z2k(y Cki Vunknot), where the signs Cki = 1 are

determined by the above construction. Since unknot = 1, plainly VK = k z 2k i eki-

Note that this procedure shows that VK is indeed a polynomial (rather than a power

series) for any knot K; a similar argument works for links.

Now let L be a link, and suppose that L' is obtained from L by tying a knot K

locally on one of the components. We can echo the above construction, expressing VL' as

>Ck z2k i ki VL, where the signs eki are same as above. Thus' VL = LVK. It follows
that the power series
7 IN VL( )
S K, (2) Z VKm (Z)
where K1,..., Km are the components of L, is invariant under PL isotopy of L. Note

that the above formula can be rewritten as V17 VL (VKI ". VK, 1)V1, mean-

while VK, "" Km is of the form 1 + b1z2 + -. + bz2" for some n, where bi(L) =

i+...+i ci(K1) .. cim (K,). We find that V1 is of the form

V/(z) z- (ao + aZ2 + a2 +...),



1 The conclusion is, of course, well-known, cf. [20], but we need the argument in order to
set up notation for use in the proof of Theorem 3.4.









where a, = c (a_ilbl + + (,,1' ). Hence V1 c Z[[z]] (rather than just Q[[z'l]]), and

Ck ck (mod gcd(co,... Ck-1)). In particular, for m = 2 we see that co(L) = co(L) is the
linking number, and ac(L) = cl(L) co(L)(cl(Ki) + cl(K2)) is (cf. [22]) the generalized

Sato-Levine invariant!

Under the generalized Sato-Levine invariant we mean the invariant that emerged

in the work of Polyak-Viro (see [2]), Kirk-Livingston [16, 22], Akhmetiev (see [1]) and

N I.: ,i-ii-Ohyama [30]; see also [25].

Remark. For m = 2 we can also obtain a (L) from cl(L) by applying the projection

+ from Remark (i) in 2, with a certain choice of B\ (cf. [16]). This is not surprising,

because Cl has KL-type 2 by Proposition 3.1, hence N(cl) is invariant under PL-isotopy by

Remark (ii) in 2.

Remark. Clearly, the power series V7 is actually a polynomial if the components of L are

unknotted or, more generally, have no non-local knots. Due to a splitting of the multi-

variable Alexander polynomial (see 4), V7 splits into a product of a polynomial (which is

a quotient of the original VL) and a power series, both invariant under PL isotopy.

Theorem 3.2. (i) For each L: mS1 c S3 and 1,':1 n E N there exists an F, > 0 such that

if L': mS1 S3 is CO E-close to L,


V,(z) 7V(z) mod (z").

(ii) V1 can be ;,,.:,;,;. 1;, extended to all tol'. I1.:. 'l links in S3, preserving (i).
(iii) The extended V1 is invariant under TOP isote'i;, of L.

Of course, the extended V* need not be a rational power series for some wild links

(which therefore will not be TOP isotopic to any PL links).

Proof. The coefficient dk of V1 at zk is of (monochromatic) type k since it is a polynomial

in the coefficients of the Conway polynomials of L and its components, homogeneous

of degree k with respect to the degrees (in z) of the corresponding terms. Since dk is

invariant under PL isotopy, by Theorem 2.2 it is invariant under k-quasi-isotopy. The








assertions (i) and (ii) now follow from Theorem 1.3(a), and (iii) either from [25] or,

alternatively, from (ii) and compactness of the unit interval. E

Proposition 3.1 implies that each ak(L) is of KL-type 2k. It is easy to check that

aI(L) = cl(L) co(L) (i cl(Ki)) is, in fact, of KL-type 1 and that


a2(L) 2(L) (L) (( c(KL))) co(L) c2(K ) + Yc + c(Ki)c1(K)

is of KL-type 3.

Proposition 3.3. a2(L) is not of KL-type 2 form = 2.

Proof. Consider the case lk(L) = 0, then 2(L) = c2(L) cl(L)(ci(Ki) + ci(K2)), which
can be also written as as(L) a3(L)(a2(Ki) + a2(K2)), where ai(L) denotes the coefficient

of VL at z'. The "third d. i .- ,i v.-, of ci2(L), i.e., restriction of a2(L) to M3 can be found
using the Leibniz rule (XV)(L,) = x(L+)(Ls) + X(L,) (L_). In the case where the 3
singular points are all on the same component Ksss, it is given by

a2(Lsss) = a2(Looo) a2(L++o)ao(Koo-) a2(+o+)ao(Koo) a2(Lo++)ao(K-oo)

where, as usual, + and stand for the overpass and the underpass, and 0 for the smooth-
ing of the crossing s. Assuming that each L** on the right hand side has three compo-
nents and each K,, only one component, we can simplify this as co(Looo) co(L++o) -

co(Lo+o) co(Loo+).
Let o: S1 c- S2 be a generic C1-approximation with 3 double points of the clockwise
double cover S1 S1 C R2, and let A, B, C, D denote the 4 bounded components of

R2 \ c(S1) such that D contains the origin. Let Ks, e M31 be the composition of p and
the inclusion R2 C S2 c S3, and let K: S1 S3\K.ss(S1) be a knot in the complement of

Kss linking the clockwise oriented boundaries of A, B, C, D with linking numbers a, b, c, d
such that a + b + c + 2d= 0. Finally, define Lss, E LM2 to be the union of K and Kss,
then Ik(L.ss) a + b + c + 2d 0 and we find that

a2(Lss) = (a + b + c+ d)d (a + c+ d)(b + d) (a + b + d)(c + d) (b + c+ d)(a + d).







14

This expression is nonzero, e.g., for a = 1, b = 2, c = -3, d = 0. E

By Proposition 3.1 and Theorem 2.2, a,(L) is invariant under 2n-quasi-isotopy.

However, according to Proposition 3.3, the following strengthening of this assertion cannot

be obtained by means of Theorem 2.2.

Theorem 3.4. an(L) is invariant under n-quasi-isote'ii

The proof makes use of the following notion. We define colored link homote('i':

to be the equivalence relation on the set of links colored with m colors, generated by

intersections between components of the same color (including self-intersections) and

addition of trivial component of any color, separated from the link by an embedded

sphere. Thus an m-component colored link L is colored link homotopic with an (m + k)-

component colored link L' iff L' is homotopic to L U Tk through colored link maps, where

Tk is the k-component unlink, split from L by an embedded sphere and colored in some

way. Such a homotopy will be called a colored link homotopy between L and L'.

P -. Let us start by considering the above procedure for computing the Conway

polynomial of a knot K in more detail. One step of this procedure yields Ck(K) =

ck(unknot) + E Cick(Li) and ck(Li) = EijCk- l(Kyi). Since co = 1 for every knot, Ck(K)

can be computed in k steps, regardless of the number of crossings in the diagram of K.

Moreover, if one is only interested in finding Ck(K) for a given k (which would not allow,

e.g., to conclude that VK is a polynomial not just a power series), the computation could

be based on arbitrary generic PL homotopies rather than those -ir--i.- I 1 by the diagram

of K. In particular, we allow (using that co = 0 for every link with > 1 components and

each ci = 0 for a trivial link with > 1 components) self-intersections of components in the

homotopies from Li's to split links, so that Kij's may have three and, inductively, any odd

number of components. For such an n step procedure, the equality VK = k,, z2k 1 i kiT

where the signs eki are determined by the homotopies, holds up to degree 2n.

Now let L be an m-component link, and suppose that L' is obtained from L by a

generic PL link homotopy H with a single singular level L,. Color the components of L

with distinct colors, then the (m + 1)-component smoothing Lo of Ls is naturally colored

with m colors. Suppose inductively that Loi2...i, is one of the smoothed singular links









in a generic PL colored link homotopy Ho0i...i between the link Loi2...i and some link

L', which either has one less component than L,, or coincides (geometrically) with

L. If there exist such homotopies Hoi2...i0 for 1 = 1,...,2k, we iv that L8 is a +(-)-

skein k-quasi-embedding, provided that the self-intersection in H is positive (negative).

A skein k-quasi-isote'l' is a link homotopy where every singular level is a + or --skein

k-quasi-embedding, depending on the sign of the self-intersection.

To see that a, is invariant under skein n-quasi-isotopy 2 we return to the orig-

inal link homotopy H and denote by K and K' the components of L and L' each

concolor with two components of Lo. As in the above argument for knots, we have

VK = Ekn Z2k(iEki ) + z2n+1P and z- L kn 2k i ki1-m L) + z2n+lQ,

where P(z) and Q(z) are some polynomials, and the signs Cki are determined by the

homotopies Hoi2... Then

ZI-mVL z1- -LR + z2n+lp P zI VL 2n+
VK' VKR + z2n+lQ VK

where R(z) = YEkn z2k Zi 6ki and S(z) is some power series, and the assertion follows.

To complete the proof, we show that n-quasi-isotopy implies skein n-quasi-isotopy.

It suffices to consider a crossing change on a component K of L, satisfying the definition

of n-quasi-isotopy. Let f, Jo, ... iJ and Po,... Pn be as in the definition of n-quasi-

embedding; we can assume that P1 contains a regular neighborhood of f(Jo) containing

L(Jo). We associate to every link Lo,, such that L \ K is a geometric sublink of Lo,, but

L itself is not, the collection of positive integers d(Lo) = (do, ... d,), where di for i > 0

(resp. i = 0) is the minimal number such that the ith component of Lo, not in L \ K is

null-homotopic in Pd, (resp. is homotopic to K with support in Pd). It is easy to see that



2 By the proof of Theorem 4.9 below, and since ca,2n and a2n,0 in Theorem 4.8 depend
only on the linking number, for n > 0 each a, is actually invariant under skein (n )-
quasi-isotopy.









d(Loi2...i +) is obtained from d(Loi ...i) by one of the following two operations:

(do,...,di,...,dm) (do,... ,di + ,di + ,... ,d );

(do,..., d ,..., dj,..., dm) (do,..., m ax(d d ),. d ,. dm).

Conversely, if for some i the operation ai and all operations 3ij 's lead to collections of

integers not exceeding n, one can construct a homotopy Ho0i...i between Loi2...i and some

L", which either has one less component than L,, or coincides with L.
Suppose that none of the integers d(Loi2...i,) exceeds 1 + 1, and at least r of them do

not exceed 1. If r > 1, let di be one of these r, then none of the integers 7(d(Loi ...i )),

where 7 is ai or /,i, exceeds 1 + 1, and at least r 1 of them do not exceed 1. Thus if at

least two components of Loi2...i are not in L\K and none of the integers d(Loi2...i) exceeds

1, we can construct a homotopy Ho0i...i as above, and for each singular link Loi2...ik+ in

this homotopy also a homotopy Hoi2,...i as above, so that for each singular link L,, 2

in this homotopy, none of the integers d(Loi2...i+2) exceeds 1 + 1. But it is indeed the

case for k = 1, hence for any odd k, that at least two components of Loi2...ik are not in

L\K. O

Remark. Note that the above argument does not work for geometric k-equivalence (see

end of 2) in place of k-quasi-isotopy.

Since Ck = ak (mod gcd(co,..., Ck-1)), Theorem 3.4 implies the case 1 1, and also

the 2-component case of

Theorem 3.5. [26] Set A = -1) -1)]. The residue class of CA+k modulo gcd of

CA, ..., CA+k-1 and all p-invariants of length < I is invariant under (L[ + k)-quasi-
isotetl'r

(Here [x] = nifx E [n n + ), and Lx] = n ifx E (n ,n+ ] for n Z.)

One special case not covered by Theorem 3.4 asserts that for 3-component links the

residue class of Ck modulo the greatest common divisor Ak of all p-invariants of length
< k + 1 is invariant under [L -quasi-isotopy. Naturally, one could wonder whether an

integer invariant of [L -quasi-isotopy of 3-component links, congruent to Ck (mod Ak), can
2~YCD- D~~ IV~IIVII~II~)~I~UI~~ I









be found among coefficients of rational functions in V* of the link and its two-component

sublinks. It turns out that this is not the casealready for k = 1. Indeed, we would get an

integer link homotopy invariant of (monochromatic) finite type (specifically, of type 4),

which is not a function of the pairwise linking numbers (since cl p(123)2 (mod Ai), cf.

[18, 23]). But this is impossible for 3-component links [24].

Alternatively, one can argue directly as follows. Consider the invariant 7(L) :

at(L) E ao(Lo)ai(Li), where the summation is over all ordered pairs (Lo, L) of

distinct 2-component sublinks of L. (Recall that in the 2-component case ao(L) and al(L)

coincide with the linking number and the generalized Sato-Levine invariant, respectively.)

It can be easily verified that 7(L) jumps by


p(12) ((1, 3+)(2, 3-) + (1, 3-(2, )+(, )(23+))

on any singular link L, K1 U K2 U K, with smoothing K1 U K U K3+ U K3 One

can check that this jump cannot be cancelled by the jump of any polynomial expression in

co(L) and the coefficients of the Conway polynomials of the sublinks of L, homogeneous of

degree 4.

Remark. An integer invariant of link homotopy of 3-component links is given by qA1 + r,

where q is any polynomial in 1JI) and r -p (123)2 (mod A1), 0 < r < A1.
Al














CHAPTER 4
THE MULTI-VARIABLE ALEXANDER POLYNOMIAL

The Conway polynomial is equivalent to the monochromatic case of the Co i' ,.';

potential function QL of the colored link L, namely VL(x x-1) = (x X- 1)QL() for
monochromatic L. For a link L colored with n colors, 2L E Z[x"1,. ,1, X] (the ring of

Laurent polynomials) if L has more than one component; otherwise QL belongs to the

fractional ideal (x x-1)-1 [x"1] in the field of fractions of Z[x-1].

fQL(XI,... X,) is a normalized version of the sign-refined Alexander polynomial

AL(tl,..., tn), which is well-defined up to multiplication by monomials tl ... t^. If K
is a knot, (x x-1)QK(x) = AK(x2), whereas for a link L with m > 1 components,

QL(x1,...,Xn) = X ... "AL(x(,.L..,Xc(n)), where c: {1,...,m} {1,...,n} is the
coloring, and the integers A, A1,..., AT are uniquely determined by the symmetry relation

QL(x1,... ,Xn) = (--1)mL(xi ,... ,x~'). We refer to Hartley [11], Traldi [35] and Turaev
[36] for definition of the sign-refined Alexander polynomial and discussion of the Conway

potential function.

Remark (not used in the sequel). The .,-vmmetry of the Alexander polynomial, which

forces one to work with its symmetrized version fL, goes back to the .i-vii. l. I ry of

presentations of the Alexander module or, equivalently, of the Fox differential calculus.

The Leibniz rule for the Fox derivative (restricted to the group elements)


D(fg) = D(f) + fD(g)

arises geometrically from considering lifts of the generators of the fundamental group

of the link complement to the universal abelian cover, starting at a fixed lift p of the

basepoint. If we wish to base the whole theory on the lifts ending at p, we will have to

change the Leibniz rule to

D(fg)= D(f)g- + D(g).









The "symmetric" Leibniz rule


D(fg) = D(f)g-1 + fD(g) + (f-'g-1 g-f-1)

does not seem to have a clear geometric meaning. (Note the analogy with the three first-

order finite differences.) However, it does correspond to a "symmetric" version of the

Magnus expansion, which will be behind the scenes in this section. Indeed, its abelian

version gives rise, in Theorem 4.2 below, to a power series SL, in the same way as the

abelian version of the usual Magnus expansion leads to Traldi's parametrization of QL,

discussed in the proof of Theorem 4.8 below.

Lemma 4.1. (compare [15]) (a) L(x1, X) QL(-X1 ,... ,- 1).

(b) If L has m > 1 components, the total degree of every nonzero term of QL is
congruent to m (mod 2).

(c) If L has > 1 components, the xi-degree of every nonzero term of QL is congruent
mod2 to the number ki of components of color i plus li := lk(K, K'), where K runs over

all components of color i and K' over those of other colors.

Proof. It is well-known that, if all components have distinct colors, the integer Ai from the

above formula relating QL to the Alexander polynomial is not congruent to li mod 2 (cf.

[35]). This proves (c), which implies (b). For links with > 1 components (a) follows from

(b) and from the symmetry relation in the definition of QL; for knots -from the relation of

QL(Z) with the Conway polynomial. O

Computation of QL is much harder than that of VL, for the skein relation (1) is no

longer valid for arbitrary crossing changes. It survives, in the form


-L+ L (Xi 7-1)-)Lo, (2)

in the case where both strands involved in the intersection are of color i. There are other

formulas for potential functions of links related by local moves, which suffice for evaluation

of Q on all links; see references in Murakami [29].

Dynnikov [6] and H. Murakami [29] noticed that the coefficients of the power series

QL(eh /2, .. ., C /2) are (monochromatic) finite type invariants of L.








Theorem 4.2. For a link L colored with n colors there exists a unique power series

1SL E Q[[zi,... z]] (unless L is a knot, in which case 3L E z-'Z[z]) such that

OL(X1 X1 1,... Xn- X ) = L(x, ... ,Xn).

The coefficient of 1L at a term of total degree n is of ';. n + 1; moreover, 13L './.:'/f the
skein relation

3L 1L -L= ZiLo

for intersections between components of color i. Furthermore, the total degree of ev-

ery nonzero term of ZL is congruent mod2 to the number of components of L, and

UL(4yi,... ,/ ) E Z[[yi,... y]] (unless L is a knot).

For n = 1 we see that ZSL(z) coincides with the (finite) polynomial VL(z).

Proof. Let x(z) denote the power series in z, obtained by expanding the radical in either

of x(z) = + 1 + z2/4 + z/2 by the formula (1 + t)r 1 + rt + rt2 + .... The Galois

group of the quadratic equation z = x x- acts on its roots by x -x-1, so Lemma
4.1(a) implies that the coefficients of the power series


UL(ZI,... ,Z.) := -L(X(Zl),...X ,X(Z))

are independent of the choice of the root x.

The second assertion follows by Murakami's proof [29] of the result mentioned above,

and the third from the skein relation (2). Next, since x-1 x = -(x x-),


UL(-ZI,...,-z,)= L(X(Zl)-1, ..,x(z,)-1)

= (-)m l(x(l), ,x(,))= (-)m'l (I, ... n,),

which proves the assertion on total degrees of nonzero terms.

The final assertion may be not obvious (since (k + 1) { (2- 1) for k = 1,3, 7) from

(t 4+( )k (2k t)y.
(1+4y)1/2 +4Z k + t k 1Nk
k 0









However, this power series does have integer coefficients, since this is the case for

2k 1
(l+4y)-1/2- + 2 )k(2- k k )k.
k= /

(In fact, we see that (k + 1) | 2(2k-1), and x(y) 1 (mod 2).) D

Let us consider the case of two colors in more detail. Let ci denote the coefficient

of 3L(zl, Z2) at ziz2. Since 63(z, z) = -V(z), coo coincides with the linking number co
in the 2-component case and is zero otherwise. The skein relation (2) implies that for a

3-component link K1 U K' U K2, colored as indicated by subscript, c0o coincides with

lk(Ki, KI)(lk(Ki, K2) + lk(K,, K2)) up to a type 0 invariant of the colored link. The latter

is identically zero by the connected sum1 formula

OL#iL'- (xj X-1 (3)

which follows from the definition of OL in Har'!.- [11]. Since col = co clo or by an

analogous argument, col = lk(K1, K2) lk(K/, K2). It follows that for 2-component links

cl coincides with the generalized Sato-Levine invariant a, up to a KL-type 0 invariant.

Since a, cl is of type 3, it has to be a degree 3 polynomial in the linking number, which

turns out to be T(lk3 k). (Since Ounlink 0 and fHopf link 1, the second Conway

identity [11, 15] can be used to evaluate OL on a series of links 'H-, with IkH-, n.) The

half-integer cl is thus the unoriented generalized Sato-Levine invariant, which is implicit

in the second paragraph of Kirk and Livingston [16]. Specifically, 2c1n/lk2 is (unless

1k = 0) the Casson-Walker invariant of the Q-homology sphere obtained by 0-surgery

on the components of the link; in fact, cn(L) = a()+l(L') where L' denotes the link

obtained from L by reversing the orientation of one of the components [16].

The preceding paragraph implies the first part of the following



1 The multi-valued operation of Hashizume connected sum L#jL' is defined as follows.
Let L'(mS1) be split from L(mS1) by an embedded S2, and b: Ix I l S3 be a band meet-
ing the embedded S2 in an arc, and L(mS1) (resp. L'(mS1)) in an arc of color i, identified
with b(I x {0}) (resp. b(I x {1})). Then L#iL'(mS') ((L(rmS) U L'(mS')) \ b(I x 01)) U
b(0I x I).








Theorem 4.3. Consider the power series


/, SL(zl, ..., z,)
V (zc(1)) V- (zc())

where K1,..., Km denote the components of L. Let cij denote the coefficient of 3*L(zi, z2)
at z'z'. For 2-component links

(i) aoo = lk, and all is the unoriented generalized Sato-Levine invariant, which
assumes all half-integer values;

(ii) Oi,2k-1 and (. -1,1 coincide, when multiplied by (-1) k+1, with Cochran's [5/
derived invariants P3, whenever the latter are 1. fI,' (i.e., 1k = 0);

(iii) aij is of type i + j; when i + j is odd, ij = 0.

The last part is an immediate corollary of Theorem 4.2.

Proof of (ii). It follows from [5] that C i / (-z2i iL= L(2), where rTL is Kojima's

y-function and z = y y-1. On the other hand, it follows from [13] that

(1 y2)(1 y-)Q (1, y)
VK2( -y 1)

where Q' is given by QL(X, X2) = (X1 x-'1) 2 -1) (Xl, X2) for 2-component links
with 1k = 0. It is shown in Jin [13] (see also proof of Theorem 4.8(ii) below) that under
this assumption Q' is a Laurent polynomial (rather than just a rational function). Define
3' by UL(zi, z2) z IZ2 LZI, Z2) for 2-component links with 1k = 0, then

S z 2'/(0, z)
(-1) z Vi() '
i 7K (z)
and consequently
-( -1)fi+21z2i L(O, z)
(O K,1 (0)VK, (z)
According to (i), the sign must be positive. E

It turns out that each coefficient of the power series 1 can be canonically split into a
Q-linear combination of the coefficients of certain 2"-1 polynomials.







23

Theorem 4.4. The Com, ri;, potential function of ,;,1 colored link L with > 1 components

can be ;,,:.,;,. l;i written in the form


QL(x1,..., Xn) {= "i } i1...i2k ({xi},..., {x }) (4)
l il<...
for some Pil...i2k e {Z[ 1,' .I.., ,z], where {f(xl,... x)} denotes f(xl,..., x,) +

f(-x1 .,..., -x1 ) for ,;; function f(xl,... ,x,). Moreover, if n = 21, the coefficients of

P12...n are integer.

Here is the case n = 2 in more detail:


Q(x, y) 2P(x x-, y y-) + (xy-1 + x-y)P12(x- x-,y y-).

This case was essentially known in 1986; indeed it is equivalent (see formula (*) below) to

Kidwell's decomposition [15]:


Q(x, y) = Ki(x x- y y-1) + (xy + x- y-1)K2(x X-_, y y-1).

However, the proof of Theorem 4.8(iii) below (on p-invariants) breaks down for K1 and K2

in place of 2P and P12. Moreover, the assertion on integrality of 2Pi, -2 in Theorem 4.4

will not hold already for n = 3 (resp. n = 5) if {I- 1} is replaced with {xl, x2,
Xz2
(resp. with { }) in (4), at least for some Laurent polynomial Q satisfying the
+1
conclusion of Lemma 4.1(a). (Lemma 4.1(a) is the only property of fL used in the proof

of Theorem 4.4.)

Proof. By Lemma 4.1(a), fL includes together with every term Ax"' .. xTP the term

(-1)P1+---+PAxiPI ... xnn and so can be written as a Z-linear combination of the L-

polynomials {xl ... x }. The formula


{xiM} {x 1M} = {x}{M}, (**)

where M is any monomial in x 1,..., can be verified directly (separately for M of

odd and even total degrees), and allows to express each {x1 ... xn } in the form


1 i { X 1 ( x ) 1) i}, {x,})k
lii<...








for some P.i EZ Z[i, ., z], k > 0, and some P' C [E -[:;, .. z]. The summands

corresponding to k = 1 can be included in P', and one can get rid of the summands

corresponding to odd k > 3 by repeated use of the formula


2{xx 'M}= {x-}{xf M} + {x '}{xM} + {xjxj}{M},


combined with (*). Note that (**) follows immediately from (*) and its analogue


{xiXjM} + {x 'x M} = {xXj}{M}.


But it is not clear from this approach that the resulting polynomials have half-integer

coefficients. To see this, represent 2{xijx 2Xi3 ... x2 i2k+} as


{ I r -- I X7 I I ... '71 1Y' \
{xlx2x i3 Xi2+l } 7 l 22k+l IXJ
2k
-- r--1 -1 .)3 1 i2k
+j 1 1 i ij +l -Xj+3 72 Xfe+l
j-1


Then formula (*) yields:


I21Xi3 X- XI2 I


2k+1
-1 j+ i ir i -- 1 r 1 i 1 i 1
( Jl{x }{x lx x ... (x -2X-ij IX'+lXij+2X +3) ?211) 2k ~k .
j=1

Thus each P .i' 2 can be included in the polynomials P .
Thus each 21.2

It remains to verify uniqueness of (4). Suppose, by way of contradiction, that a

nontrivial expression Q(xi,..., x,) in the form of the right hand side of (4) is identically

zero. Then so is Q(xl,..., Xn,-_, x) Q(xl,... X 1, -x ), which can be rewritten as


[xn] [x x x7~I 3 1 -i2k1 2k 1 Pil*...i2k-1,n({xi},..., {xn}) 0,
1,il<...
where [f(xi,... x)] denotes f(xl,... x) f(-x 1,..., -x). Denote the left

hand side by [x,]R(xl,..., x,), then R(xl,...., x,) is identically zero. Hence so is

R(xl,..., x,_2, x-1, xn) R(1,... x_ n-2, n-l, x), which can be rewritten as


[x_-l]


--1 ," '
S {XiX Xi2k lXi }Pil...i2k,n--,n({Xl},..., {Xn}) = 0.
1 il<

(**)









Repeating this two-step procedure ['] times, we will end up with


[xi]{1}Pi...n({xi},... {xn}= 0 or [x21{}P2...n({xi}, ,{x ) = 0

according as n is even or odd. Consider, for example, the case of odd n. By symmetry,

P1..... = 0 for each i. Returning to the previous stage


[X4] i i 1 i -1 } il...i2...k,4..n({1. {Xn}) = 0,
1il1<---
we can now substitute zeroes for P23,4...n, P13,4...n, P12,4...n, and so we get P4...n 0.

Continuing to the earlier stages, we will similarly verify that each P,...i = 0. E

Since every term of, -, x x-1 has an odd degree in x, Lemma 4.1 implies

Lemma 4.5. For every nonzero term of Pil...i,

(i) the total degree is congruent mod2 to the number of components;

(ii) the zi-degree is congruent to ki + li (mod 2) iff i {i,..., i,}.

Let us turn again to the case of two-component links colored with two colors. Let d'

(resp. d"j) denote the coefficient of P(z, z2) (resp. P12 Z2)) at z'z. Substituting x(zi)

for xi as in the proof of Theorem 4.2, we get

2 2
1 ziz2 I Z2
rx2 1 + x1 2 2- + + + terms of total degree > 4.
2 4 4

Thus coo = 2(doo + d"o) and

c = 2(d', + d,) C2 2(do + d2l) + 2 = 2(dO2 + d2) +
2 4 4

By Lemma 4.5(ii), d2i2j d+1,2j+1 = 0 if the linking number is even; otherwise

di+1,2j+1 di2j = 0. On the other hand, coo = k and cn = a (1k3 lk)/12 by Theorem
4.3(i). Thus 2d"g is 1k or 0, and 2d", is 0 or the integer ac (1k3 lk)/12, according as 1k

is even or odd. Similarly, 2doo is 0 or 1k, and 2d' is the integer ac (1k3 -41k)/12 or 0,

according as 1k is even or odd. On the other hand, it follows from Proposition 2.1 that the

KL-type 0 invariant 2doo = ((-1)lk + 1) k /2 is not of finite type. Since the linking number

coo and the half-integer cl are of finite type, the integer d1 := 2(d' + d',) is also not of

finite type.









Remark. The decomposition (4) in the case n = 2 can be modified, using the identity

S (x y)2
xy-1 + x- y = -y + 2,
xy

to the form


L(, y) 2(P({x}, {y}) + P12({ {y})) + (X P12( {y}).
xy

Lemma 4.5(ii) implies that each coefficient of P + P12 coincides with the corresponding

coefficient of either P or P12, depending on the parity of lk(L). Since lk(L) is the constant

term of 2(P + P12), the remainder (xy)-l(x y)2P12 is redundant, in the sense that it

contains no additional information with respect to 2(P + P12).

More generally, we have

Theorem 4.6. Let Pi, ..., i2 be the p.'l'. I,;;,',.*l of the colored link L, /. I,.. l in (4). The

" ,;;,,. .;;,,d 7V L, f .u, ,l by


VL(zI,...,z ) := 2 Pi ..., (z ,..., ) e Z[zi,..., zn]
l
if L has at least two components, and by VL(z) = z-1VL(z) E z-'Z[z] if n 1, determines,

and is determined by, the Co ,i, ia ; potential function QL (X, ,xT,). The coefficient of VL

at a term of total degree d is of colored type d + 1, and vanishes unless d is congruent mod2

to the number of components of L.


Proof. By Theorem 4.4, 2L determines each Pi~..., hence VL. Conversely, Lemma 4.5(ii)

implies that the coefficients of each Pi~...,i2k hence of QL, are determined by VL. The last

two assertions follow from Lemma 4.5(i) and the skein relation (2), which can be rewritten

in the form

VL+ VL = ZiVLo


for intersections between components of color i.








Let K1,... Km denote the components of a colored link L, where m > 1. It follows

from Theorem 4.4 that the power series
0,*(n r-_L(z I,. ,an) _. 1 il
Q- (Xi,... ,Xn) CK 1 (7)[[ L(,.X.. ,a) 1]
VKi (Xc(i) Xc(1)) VKr (Xc(mn) c(m)

can be expressed in the form

-(x ,... ,x ) { }il ({xl},.. x }),
Sil<..
where each P., E Z[[zi,..., z,]], and the proof of Theorem 4.4 shows that such an

expression is unique. The doubled sum of all these polynomials is, of course, nothing but

*(z1, i,... z,)
V2(Z1,... ,Zn) :=
S : VK, (Zc(1)) VKm (Zc(mn))

which contains the same information as QT by the proof of Theorem 4.6.

Corollary 4.7. (i) For each L: mS1 S3 and ,;1 k E N there exists an Ek > 0 such

that if L': mS1 S3 is CO Ek-close to L,


V, = V + terms of total degree > k,

3*, = 1* + terms of total degree > k.

(ii) V and 1* can be ;i,', '.:. 1; extended to all TOP links in S3, preserving (i).
(iii) The extended VL and S5* are invariant under TOP isotei,; of L.

Proof. As in the above argument for V1, it is easy to see that VL, as well as S, are

invariant under PL isotopy (cf. [35]). By Theorems 4.2 and 4.6, the coefficients of VL and

13 at terms of total degree k are of colored type k + 1, hence by Theorem 2.2 they are
invariant under (k + 1)-quasi-isotopy. The rest of the proof is as in Theorem 3.2. D

Remark. If the components of L are unknotted, or more generally have no non-local knots,

VL is a (finite) polynomial. Actually, VL splits into a product of m one-variable power
series and a polynomial (namely, the product of the irreducible factors of VL, involving

more than one variable), so that each of them is individually invariant under PL isotopy;









compare [35, 32]. There is, of course, such a splitting for every Pi...i producing a

plethora of PL isotopy invariants.

Lack of an analogue of Lemma 4.5(ii) for the power series 1L hinders establishing a

simple relation between the coefficients of 1L and those of Traldi's expansion of QL [35],

which prevents one from expressing the finite type rationals yij (discussed in Theorem 4.3)

as lifting of p-invariants. However, each cij can be split into a linear combination (whose

coefficients depend on the linking number) of the integer coefficients of the power series

VL, which do admit such an expression.

Theorem 4.8. Let Sij denote the coefficient of VL at zIz For 2-component links

(i) 6oo = Ik, and 6 is an integer 1':f1.:',, of the Sato-Levine invariant p(1122), but not

a finite ';I'p. invariant;

(ii) 61,2k-1 and 62k-1,1 are Cochran's [5/ derived invariants (-1) +10 whenever the

latter are / I/., l (i.e., 1k = 0);

(iii) 6ij is an integer '.:fl.:.', of Milnor's i.'] invariant (-1)j (1. .12... _2), provided
i+1 j+1
that i + j is even;

(iv) 6ij is of KL-',,I,. i + j;

(v) when i + j is odd, ij = 0;

(vi) when i + 1, j + 1 and 1k are all even or all odd, 6ij is even;

(vii) for a given L, there are only finitely i,,,.'i; pairs (i,j) such that yij 0 0 modulo the

greatest common divisor Aij of all kl 's with k < i, 1 < j and k + 1 < i + j; the congruence

can be replaced by i./;,.';:1;/ if the components of L are unknotted.

The last four parts are immediate consequences of Theorem 4.6; Lemma 4.5(i);

Lemma 4.5(ii) and integrality of the coefficients of P,*2; the definition of VL as a rational

power series. The first part follows from the discussion after Lemma 4.5.

Remark. The geometry of the relationship between p-invariants and the multi-variable

Alexander polynomial is now better understood [17]. If L is the link closure of a string

link f: {1,..., m} x I R2 x I,


AL AjF,








where AL denotes the usual Alexander polynomial, A5 the Alexander polynomial of
the string link [19], i.e., the Reidemeister torsion of the based chain complex of the pair
(X, X n R2 x {0}), where X = R2 x I \ im and Fe a certain rational power series,
determined by the p-invariants of f.

Proof of (iii). Since we are only interested in the residue class of every 6ij mod Aij, we
may consider the polynomial VL 2(P+P12) in place of the power series L = 2(P*+P*2).
By Theorem 4.4 and Lemma 4.5(ii), 2L is uniquely expressible in the form

({Xl}{X2})2 L(Xl, X2) Ql1({}2, x2}2) + {X12-1}{XI}{1X2}Q({XI}2, {2}2)

for some Q1,Q2 C Z[i,Z21, where A = 0 or 1 according as lk(L) is odd or even. Set

y, = x2 1, then x-2 = 1 Yi + y. y + .... The identities



{xjXx}{xIX} {2 (X-2 + -2)(x )(X 1) (2 yI-y2 + Y+Y )Y12

allow to express (x1x2)-AL(X1, x2) as a power series TL(yI, Y2) with integer coefficients.
Let us study this substitution more carefully. Let d.j, d' denote the coefficients
at z'z' in P and P12, and let us write (k,l) < (i,j) if k < i and / < j. Then the
coefficient ej at yy in the power series R2, defined by the equality (yIy2) -'R2(yi, Y2)

(ziz2)A-1Pl2(zl, z2), is given by

e4 (-1)(i-k)+ (j-1)d- (-_)'+j k
(k,l) (ij) (k,l) (ij)
(the latter equality uses that d"' = 0 if k + 1 is odd), and similarly for the coefficients ei of
the power series R1, defined by (yiy2)Ri(yi, Y2) = (Z1i2)P(Z1I z2). Now the coefficients

eij of TL(y, y2) are given by

ey 2e + 2e%' + (- )-k + (-1)j-1 t
k 2e- + 2e' + (- )+ > ((i k) + (j )>/',
(k,1)<(ij)









The key observation here is that all coefficients on the right hand side are even if i + j is

even.

Let us consider the case A = 1. Then by Lemma 4.5(ii), d' = 0 unless both i and j

are even, and dj 0 unless both i and j are odd. Hence e% = 0 modulo gcd{e" (k, 1) <

(i,j)}, unless both i and j are even, and similarly for e Now it follows by induction that

eij 2e' 2d' or eij 2e.j = 2dj modulo Eij : gcd{ek | (k,1) < (i,j)} according

as i and j are both even or both odd. Thus ej 2(dj + d%) (mod Ejy) if i + j is even.

Clearly, the latter assertion holds in the case A = 0 as well, which can be proved by the

same argument.

Finally, since -A Ik k-1 (mod 2), and xn 2 are expressible as power series in yi

with integer coefficients and constant term 1, the coefficients eij of Traldi's power series

TL(yl, y2) (X2)lk-l IL are related to eij by congruence modEyi. This completes the
proof, since by Traldi [35], each eij is an integer lifting of (-l) +1'(1...12... 2).
i+1 j+i


Remark. The above argument yields a new proof of another result due to Traldi:

2p(1...12... 2) 0 when i + j is odd [34].
i+1 j+i
Proof of (ii). We recall that p(1... 12) and p(12... 2) identically vanish, with the excep-

tion of p(12) Ik [28]. Hence by (iii), o0,2k 62k,0 0 (mod 1k) for each k. So if Ik = 0,

every nonzero term of either P* or P,2 involves both zl and z2. (Alternatively, this follows

from Jin's lemma mentioned in the proof of Theorem 4.3(ii).) By Lemma 4.5(ii), every

nonzero term of P2 has to further include each of them once again, i.e., P*2 is divisible

by z1z2. Hence, firstly, 62k-1,1 coincides with the coefficient of 2P*(zi, z2) at z k 2, and,

secondly, this coefficient is not affected by adding (x(zi)x(z2)-1 + x(z2)x(x)- 1)P2(i, z2)

to 2P*(zl, z2), where x(z) is as in the proof of Theorem 4.2. O

Theorem 4.9. The coefficient of V at z ... z is invariant under k-quasi-isote':,, if

max(i,..., in) < 2k.

Proof. This is analogous to Theorem 3.4. One only needs to show that the coefficient

in question is invariant under restricted skein k-quasi-isotopy, where a restricted -skein









k-quasi-embedding is defined as before, but with additional restriction that each Loi2...,ik

includes L \ K and each Ho2... i is fixed on L \ K. This is done by the same argument,

replacing every occurrence of the polynomial zI-mVL with VL. E


Corollary 4.10. (a) [26] Cochran's invariants /3 are invariant under k-quasi-isot(i',,

(b) Milnor's invariants p(1 ... 12... 2) of even length are invariant under k-quasi-

isot(',,i if each index occurs at most 2k + 1 times.

Part (b) covers (and largely improves) the corresponding case of

Theorem 4.11. [261 All p-invariants of length < 2k + 3 are invariant under k-quasi-

isot ,i,,i














CHAPTER 5
THE HOMFLY AND KAUFFMAN POLYNOMIALS

We recall that the HOMFLY(PT) P y. 1,;;,,..',,:l and the Dubrovnik version of the

Kauffman F ''.;; ,,ii.:,,l1 are the unique Laurent polynomials HL, FL cE Z[x1, y-1] satisfying

Hunknot = unknot = 1 and

XCHL- x-1HL = yHLO,

XFL -- X-FL Y(FLo xWw(L)-u(FLo),

where L+, L_, Lo and L8 are as in the definition of the Conway polynomial (cf. 3), L, is

obtained by changing the orientation of the iight" of the two loops in L8 (corresponding

to either the two intersecting components or the two lobes of the singular component)

and oriented smoothing of the crossing of the obtained singular link L', and w(L) denotes

the writhe of the diagram of L, i.e., the number of positive crossings minus the number of

negative crossings (so that w(L+) 1 = w(L_) + 1 = w(Lo)). The versions of HL and FL

in Lickorish [20] are obtained as HL(-ia, iz) and (-1)"-1FL(-ia, iz).

Theorem 5.1. Let et denote the (formal) power series 7 and consider the power

series
HL FL
HL := and FL :=
HK1 ... HK and FK1 ... FK,
where K, Km denote the components of the link L.

(i) For each L: mS1 c S3 and 1,;1 n E N there exists an F,, > 0 such that if

L': mS1 c_ S3 is CO E-close to L,

H,(eh/2, e/2 -h/2) H(ech/2,eh/2 -h/2) mod (he),

FL,((c-1)h2, eh/2 --h/2) F (C(c-1)h/2, h/2 -h/2) mod (h).


(ii) HL and FL can be ;,,::.,;,. 1: extended to all TOP links in S3, preserving (i).

(iii) The extended H1 and FL are invariant under TOP isote'i' of L.









Proof. The connected sum formulae for HL and FL [20] imply that H* and F* are

invariant under PL isotopy. (Note that the connected sum in Lickorish [20] is Hashizume's,

not the componentwise connected sum of Melikhov and Repovs [25].) On the other hand,

it was noticed in Lieberum [21] (compare [9, 4, 33]) that the coefficients of the power series
oo k+m-1
HL(ech/2, C h/2 -h/2) k pi k 1
ll( d / -h) Pki c' c Q[c]l[[hl]]
k=0 i=0
oo k+m-1
FL(e(c-1)h/2, Ch/2 2-h/2) iC Ck 1
> qkich (Q[c][[h]]
k=0 i=0

are (monochromatic) finite type invariants of L. Specifically, each pki and each qki is of

type k, moreover oi = qo = 6,m-,i (the Kronecker delta). (The argument in Lieberum [21]

was for HLHT2 and FLFT2, where T2 denotes the trivial 2-component link, but it works as

well for HL and FL, compare [3].) The rest of the proof repeats that of Theorem 3.2. O














REFERENCES


[1] P. M. Akhmetiev, D. Repovs, A generalization of the Sato-Levine invariant, in:
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Inst. Math. 221 (1998), 60-70.

[2] P. M. Akhmetiev, J. Malesic, D. Repovs, A formula for the generalized Sato-Levine
invariant, Mat. Sbornik 192 no. 1 (2001), 3-12; English transl., Sb. Math. (2001),
1-10.

[3] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472.

[4] J. S. Birman, X.-S. Lin, Knot jp ...;;i,.,.;',,,l and Vassiliev's invariants, Invent. Math.
111 (1993), 225-270.

15 T. Cochran, Geometric invariants of link cobordism, Comm. Math. Helv. 60 (1985),
291-311.

'. I. A. Dynnikov, The Alexander p y.'l;;,i',.:rl in several variables can be expressed in
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BIOGRAPHICAL SKETCH

I was born in Moscow, Russia, in 1980.

My high school was the Moscow State Fifty-Seventh School, where I studied in a

mathematically oriented class in 1992-96.

In 1996-2001 I continued my education at the Mechanics and Mathematics Depart-

ment of the Moscow State University, which conferred a bachelor's degree in pure and

applied mathematics on me.

I have been in the Graduate School of the University of Florida since 2001.