; TeX output 2005.08.28:2342sk5K`yff cmr10Acknowledgements$2K`y cmr10I?woulddliketothankbGothmysupGervisorsformanyhelpfulconversationsandUUmuchusefuladvice.2ITwouldTliketothankDr.W.B.R.LickorishparticularlyforhispainstakingcheckingUUofmywork,includingthepapGerpresentedhereaschapter6.2IwouldliketothankDr.H.MortonparticularlyforgivingmetheproblemoutS_ofwhichtheideasforthisthesisgrew,Sandforgivingmetheprogramfrom["V cmbx1024]UUwhichwasusedtocalculatethelowerbGoundsforthebraidindex.$2I(would(alsoliketothankmywifeforhersuppGortandadvicewhilewritingthisthesis,qTimAucklandforhisproGof-reading,andnumerousothersfortheirvqaluableUUcomments.2TheauthorwassuppGortedduringthisworkbytheScienceandEngineeringResearchUUCouncilofGreatBritainandbyT*rinityCollege,Cambridge.{UU5{*sPrefaceEܟ!є2AP linkPisanembGeddingofacircle,Qoradisjointunionofcircles,QinR pٓRcmr73l.pW*eusuallyhconsidertwohlinksasbGeingthesameifonecanbecontinuouslyhdeformedintoWtheother.y$F*ortechnicalreasons,XfweactuallyrequirethedeformationtobGeaUUdeformationofthewholeambientUUspaceR Rp3qū.2A4closed4braidisaspGecialsortoflink,lwhichismadebyembGeddingthecirclesintoasolidtorussuchthatthecirclesgomonotonelyroundthetorus,andUUthenembGeddingthesolidtorusinastandard(unknotted)fashionintoR Rp3qū.2The~factthateverylinkcanbGerepresentedasaclosedbraidhasbGeenknownsince 1923[1],;[7].pItisonlyrecently*,however,that thishasbGecomesuchanimpGortant featureinthedevelopmentofknottheory*.FInparticular,Mtheso-called b> cmmi10P%빫-pGolynomialAxof1985[12],Eq[17]canbeinterpretedintermsofbraids[13],Eq[34].TheY\resultingcalculationtechniquefortheP (y-pGolynomialofalink[24]hasacomplexitywhichispGolynomialinthenumbGerofcrossings,ۖbutexponentialinthenumbGerofstringsinthebraidrepresentation.ItisthereforeofinteresttorepresentUUlinksasclosedbraidswithasfewstringsaspGossible[14].2The3`minimalnumbGer3`ofstringswhichcanbeusedtorepresentagivenlinkasoaclosedbraidiscalledthebraidindexofthatlink[4],uandisaninterestinginvqariantUUofthelinkinitsownright[5],[11],[28].2The?aimofthisthesisistwo-fold.EkFirstly*,we?developamethoGdforcalculat-ingkagoGodkupperboundforthebraidindexofalink.tSecondly*,qweensurethatthismethoGdispractical,inthesensethatitcanbeusedto ndaclosedbraidpresentationUUofthelinkwhichhasthatnumbGerofstrings.2In9f[14],r0Jonesgivesclosedbraidpresentationsforallknotsupto10crossings.W*epextendthisresultto11crossingknotsandalllinksupto10crossings,aslistedUUin[8].{UU6{؍k\+- cmcsc10Prefraces2A7closedHbraidpresentationofalinkisausefultoGolfortwomainreasons.Firstly*,}auclosedbraidpresentationisausefulstartingpGointforstudyingmanypropGertiesGoflinks,}includingtheP-polynomial[22],}theDubrovnikD-polynomialor!Kau man'sF-pGolynomial[16],TotherLie-grouprelatedinvqariants![34],andHeckeUUalgebras[14].qԍ2Secondly*,itprovidesaconcisewayofwritingdownagenerallink.5WhilstthenotationinventedbyConwayin[8]iscertainlyconcisefortherangeoflinksin*thatpapGer,3itisnotclearhowtoextendthisnotationinanon-arbitrarywaytozgenerallinks.7Mostothernotationsaresimplyawayzofdescribingaplanarrepresentationofthelink,arediculttocalculatewith,andhavecomplicatedself-consistencyUUconditions.+P^4"V cmbx10SummaryThe%jdepGendencyofchaptersisthatchapters1to5needtobGereadinorder,chapter7needstobGereadafterchapter3,andchapter6isindepGendentfromtheUUrest.qThereisanindexofde nitionsattheend.2Chapter1dealswiththeconventionsandde nitionswhichwillbGefre-quently;usedintherestofthisthesis.$ThesectionsonSeifertDiagramsandPre-Diagramsrpresentnewmaterial,HwhilsttherestcanbGeskippedata rstreadingUUandreferredbacktoasnecessary*.2Chapter1"2startswiththetopGologicalaspectsoftheproblemofobtainingagoGodupperboundforthebraidindexofalink,Vstartingfromadiagram.6ThischapterUUreducestheproblemtooneingraphtheory*.2Chapter:3dealswiththeresultinggraphtheoryandprovidestheimpGortanttheoremsΔneededforecientcalculation.݅Ontheway*,,ausefulnotationforplanarhgraphsisintroGduced.Asano -shootofthiswork,ݭageneralboundforthebraidindexofalinkisdevelopGedwhichusesonlythenumbGerofcrossingsinUUadiagramofthelink.{UU7{&k\Prefraces2Chapter4usesthetheoremsfromchapter3toconstructusefulalgorithms.ThesealgorithmscalculatetherequireduppGerboundonthebraidindex,CandgiveUUaclosedbraidpresentationwiththatmanystrings.v2ChapterZ5appliesthesealgorithmstotheknotsandlinkslistedin[8],[QandcommentsUUontheresults.qTheactualresultsarepresentedintheappGendix.2ChapterzM6*canbGereadindependentlyfromtheotherchapters.(Itdealswitha1subGclassoflinks,8andusestheP-polynomialtodeterminewhenthealgorithmofRchapter4reducestheuppGerboundonthebraidindexbelowthenumbGerofSeifertUUcircuits.2Finally*,chapter7discussessomequestionsarisingfromthisthesis,provessome&theoremswhichdonot tneatlyintoanyearlierchapter,andgivessomeconjecturesUUandtopicsrequiringmoreinvestigation.'9TOutline<_TW*enowgiveashortintroGductiontothetechniques,hideasandde nitionswhichUUwillbGeusedinthisthesis.2In^ordertogetanuppGerboundforthebraidindexofalink,aRwestartwitha2theoremofY*amada[37],9whichsaysthatthebraidindexofalinkisbGoundedabGove]LbythenumbGerofSeifertcircuitsinanydiagramofthatlink.SinceaclosedubraidpresentationwithnBstringscannaturallybGepresentedasalinkdiagramNwithn-6Seifertcircuits,Ftheaimisto ndalinkdiagramwithasfewSeifertUUcircuitsaspGossible.2T*oOthisend,weinventtheideaofareducingcut.^Thisisjustawayofalteringl alinkdiagrambypickinguponepieceofthelinkandputtingitdownsomewhereWJelse,WsuchthatthenumbGerofSeifertcircuitsisreduced.wIntheorem EJffxs  *Chapter6isapapGerwhichhasbeenacceptedforpublicationintheMathe-maticalProGceedingsoftheCambridgePhilosophicalSociety*,(andshouldappearthereUUinearly1992.{UU8{ ƍk\Prefraces7.1,|weFshowthatanytwolinkdiagramsofthesamelinkarerelatedbyasequenceofcutsortheirinverses.HXW*earethereforeinterestedinsequencesofcutsortheirinversesUUwhichreducethenumbGerofSeifertcircuitsasmuchaspGossible.f}2T*o6Pmaketheproblemtractable,werestrictourattentionintwoways.Firstly*,wef\onlyconsidercuts,nottheirinverses.Secondly*,wef\onlyconsidersequencesfCwhichreducethenumbGerofSeifertcircuitsaftereachcut,~i.e.se-quences uof$': cmti10r}'educingcuts.(If uthenumbGer uofSeifertcircuitsintheoriginallinkdiagram,.S 8,.isFϱszp(S)_,andFthemaximumlengthofareducingcutsequenceisrGcs(S):f,UUthentheresultinguppGerboundforthebraidindexisszp(S)8 !", cmsy10rcs(S)D$.2Unfortunately*,(the]classofallreducingcutsisslightlytoGolargetoworkwith,YDandso(inchapter2)wede nesemi-lo}'calcuts,YDrestrictourattentiontolsemi-loGcalreducingcutssequences,andconsiderthepossiblyweakerlboundszp(S)8srGcs(S)ck.qItUUisnotknownwhetherthesebGoundsare,infact,di erent.2Semi-loGcal9cutsturnouttobeanimportantclasswithwhichtowork.hXThelemmashandtheoremswhicharecentraltothedevelopmenthavehighlyinductiveproGofs,.#and$VmostofthesewouldnotbeprovqablewithinaclasswhichwaseitherlargerUUorsmaller.2AnO-exampleofthisistheorem3.4,PhwhichwouldfailfortheslightlysmallerclassofloGcalcuts.CTheorem3.4essentiallysaysthatsemi-loGcalcutscommute,i.e.)@if}(c1|s;c2)$yis}agivensemi-loGcalreducingcutsequence,then(c2|s;c1)isalsoasemi-loGcal "reducingcutsequencewiththesamee ect./Thisisnotquitetrue,andmsomealterationsneedtobGemadetothegivencutsequencetomakeittrue,butUUitissucientlyclosetothetruthtobGeagoodintuitiveUUguide.2InJordertoproveJtheseresults,%Mwehavetoconsideramuchmorerestrictiveclass=.ofcutsthansemi-loGcalreducingcuts,u6namelyincompressiblelocalmonotonecutsi(aloGcalcutissemi-local,nandalocalmonotonecutisreducing).FItisonlyin'thissmallclassthatcutscanbGepulledapartsothattheyaredisjoint(lemma2.10),?3andmthisisnecessaryformostofthedevelopment.Havingmprovedthis,the.remainderofchapter2isdevotedtoshowingthatifthereisasemi-loGcal{UU9{ "k\Prefracesreducingicutsequenceoflengthn ԆstartingfromalinkdiagramS J,othenthereisanUUincompressibleloGcalmonotonecutsequenceoflengthn DstartingfromS .2TheiIma8jorityoftheworkneededtoprovethisisdoneintheorem2.14,whichshows(thatmonotone;semi-loGcalreducingcutsequencesgiverisetoincompress-ibleloGcalmonotonecutsequencesofthesamelength.UTheamountofworkthenrequiredtoremovethewordmonotoneiscomparativelysmall.OncethishasbGeenfestablished,wecanfreelyconvertbGetweenthesmallclassofincompressibleloGcalCJffff[$\OverUnder"cBff[[CJff2|FigureT1.2X-ff[suchUUthat<,a(i)<q1|s (n:P80)=q1 (n:P81).)D(ii)<q2|s (x8t)=tyF.2TheseconditionssayrespGectivelythattheimagesof VonthetwoendsoftheUUcylinderB^28I"arethesame,andthat ƫislevel-preserving.y82AUUclose}'d]ޫbraidisthecompGositemapn_n:P8I썑p ,UX!qƫB^2ff'ffS)8M\ElementaryBraid"$_ffSSN>ff2|FigureT1.3X-ff[(2ThisUUgivesusade nitionofthebr}'aidgroup,~9Bq(n)=h1|s;:::;n1u j(i)UUand(ii)1qʷi:e (1:3)SuppGoseUUb2Bq(n)).qThenUUwewillusehn:bitoUUdenotethelinktypGegivenbytheclosureofthebraidb.%exTheBraidIndexw(The0br}'aid^indextofalink`ܫ,writtenbi(`):,isthesmallestn suchthatthereisaclosedUUn U-stringUUbraidrepresentationofthelinktypGeof`.PTheoremT1.1SuppGoseUU`=`1St8`24NisUUasplitlink.2ThenUUbi(`)=bi(`1|s)8+bi(`2)`.ProQof Given)anni -stringclosedbraidpresentationfor`iJ(fori=1UUand26(),we]{canmakeann1S+8n2%s-stringclosedbraidpresentationof` 墫bylayingthemside-by-side,UUsobi(`)bi(`1|s)8+bi(`2):{UU18{r-Introductions2Conversely*,79given ann -stringclosedbraidpresentationof`4,considerjustthatpartofthebraidcorrespGondingto`i  .(Thisgivesanni -stringclosedbraidpresentationUUfor`i M,wheren=n1S+8n275,andtherefore a썒@bi(`)bi(`1|s)8+bi(`2)asUUrequired.ff̟ffffffff̎(؍NoteT1.2This)theoremmeansthatforthepurpGosesofcalculatingthebraidindexofalinkUU`,UUwemayassumethat` Visnotsplit.2W*eMHaimtoobtaingoGodMHboundsforbi(`),Eandto ndbi(`)HexactlyforasmanyUU` VasUUpGossible.2LowerQbGoundsareobtainablefromtwoQsources.p[Firstly*,QthelinkswithverysmallXbraidindexareobvious.0Theonly` O\withbi(`)<2+(garetheemptylink(withbraidrindex0)andtheunknot(withbraidh1:1iA).CIfbi(`)=2)v,z3then` TmustbGeeitherUUan(n;2)-torusknotwithbraid a썒ݷh2:䍐[ٮ2n11>i (n2Z 2;n6=0;1);orzelsethen -twisted2-cablelink(withmatchingorientations)oftheunknotwithUUbraida썒|h2:[ٮ2n፮1 ʷi (n2Z 2):"2Secondly*,weäcanusetheP -pGolynomialof` toobtainalowerbGoundonbi(`)[21],UU[10],[22].qW*ewillexplainthisinmoredetailinchapter4.2UppGerboundsforbi(`)arethesub8jectofmostofthisthesis.WTheapproachisacombinatorialone,tryingtochangealinkdiagramof` =intoaclosedbraidwithUUasfewstringsaspGossible.{UU19{wIntroductions,ASeifertDiagramsYȍW*e18nowde neaslightlydi erentkindoflinkdiagram,8qwhichismoreusefulforourUUpurpGoses.2AUUSeifertdiagr}'amisa5-tupleSZ=(A;E;iA;iEm;[٫)]bsuchthat,a(i)<AFandUUEareUU nite(disjoint)sets.)D(ii)<iA P:qA8S^1 ,UX!S^25.''(iii)<iE 3-:qEm8I,UX!S^2z.'GC(iv)<imG9(iA)8\im UT(iEm)=iE(Em8@8I) .*`(v)<͙:qEZ!f+;g]k.'GC(vi)<NeartoanyofthepGointsiniEm(Em80).p,thesituationisasshowninthe< rstpartof gure1.4.oNeartothepGointsiniEm(Em81).?,[thesituationis<asUUshowninthesecondpartof gure1.4.,kvff[e2iRYWuffff.UC\SdEdges"*+ffYWuff2|FigureT1.4X-ff[=2Intuitively*,awSeifertdiagramisasetofdisjointcircles,withsomedisjointsigned%>edgesbGetween%>them,Y8alwaysrunningfromananti-cloGckwisecircletoacloGckwiseUUcircle,asseenfromtheedge.2F*orOeacha2A\,ͱiA(a8S^1 C)4yiscalledaSeifertcir}'cuitث,orjustacircuit.+Foreach e2E;,iEm(e8I),%is calledane}'dge[.CThepGointsiEm(Em l1l&fes2)2arecalledvertic}'es!,{UU20{Introductionsand-maybGemarkedon gures.dIftwoedgesgobGetweenthesamepairofSeifertcircuits,UUtheyarecalledp}'arallel.qF*orUUbrevity,wemaywriteC{im(S)=im (iA)8[im UT(iEm):e (1:4)W*eUUsayS _isc}'onnected]ޫwhenUUimq(S) CisUUconnected.:2ThereEisamapfromlinkdiagramstoSeifertdiagrams,Hwhichisillustratedinthe rstpartof gure1.5.c;Eachcrossingischangedasshown,andthenthejoiningofthecrossingsinthelinkdiagramgivesthejoiningoftheSeifertcircuitsBintheSeifertdiagram.VTheverticesmarkthepGointsontheedgeswheretheUUcrossingwas.oxljff[XL$ffBff[(,U\SdMap"#ꒉff[[L$ff2|FigureT1.5X-ff[ h2This9mapisclearlyabijection,?VuptoadeformationofS^2 D,sincetheinverseisgivenbythesecondpartof gure1.5.9 ThuswecanworkwithSeifertdiagramsinsteadUUoflinkdiagrams."׍2TheUUSeifertdiagrams7SZ=(A;E;iA;iEm;[٫)andUUS^0.7areUUsaidtobGeambientisotopic whenS0T͍+3=ӎ(A;E;jA;jEm;[٫){UU21{1IntroductionsandUUthereisadeformationh mofS^2WsuchthathiA J=jA.ZIntroductions2F*orceacha2Ae,g:iA(a8S^1 C)4=iscalledapr}'e-circuitث.Itcmaycrossitself,g:otherpre-circuits._oredgesinS 㳫.dEachplacewhereapre-circuitcrossesapre-circuitisassignedhasigninthesamewayhthateachcrossinginalinkdiagramisassignedasign,<and5withthesamemeaning.gCwithoutchangingtheequivqalenceclassofP .h*Nownomatterhowthepathsarejoinedupinsideandoutsideb,asspGeci edbya 8,weUUstillendupwithatleasttwopre-circuits,noneofwhichintersect.}Wff[>cffW҉ff[VdcR\AlternateCrossings"R1ʉff[[cff2|FigureT1.8X-ff[2ThisӱproGcessreducesthenumberӱofintersectionsbetweenӱpre-circuitsandpre-circuits,andwhenceiteventuallywterminates.Sincetherearenomorein-tersections,Ԧtheresultingpre-diagramisinfactaSeifertdiagram,andhenceisambientisotopictoS Y.Ateachstage,thenumbGerofpre-circuitscouldonlyincreaseUUorremainunchanged,andsoszp(S)sz(Pc)Easrequired.5Õff̟ffffffff̎{UU26{ҍs3jChapter2.fgCuts(ܩQ7De nitionsGivenUUaSeifertdiagramS ,acutH-forS _isanembGedding=J})cq:S^1C,UX!S^2suchUUthat(܍,a(i)<TheUUimagec(S^1 )Pcontainsexactlyonevertexv gofS .)D(ii)<ApartUUfrom(i),c isingeneralpGosition.''(iii)<TheUUvertexv gliesonanedgeeV.qAtv,c crossese Rfromlefttoright.2Thevertexv ?andtheedgee karecalledtheactivevertexandtheactiveedgeforUUc0.2GivenNaSeifertdiagramS andacorrespGondingcutc), wecande neanewSeifertqdiagramST=c٫asfollows.qSm[8c#A'isapre-diagram,exceptthatc passesthrough^avertexv ofS T.ChangeSm[8c᛫nearthecrossingasshownin gure2.1.Thisgivesapre-diagram[ST=c]ofsizeszp(S)81->$.W*enowde neST=catobGetheSeifertUUdiagramcorrespGondingto[ST=c]^.ff[j^Ǘffωff[1}f7\CutCrossing",̉ff[[^Ǘff2|FigureT2.1X-ff[{UU27{Cutss2Intuitively*,>uthe8pieceofthelinkwhichwentoverthecrossingatv 5hasbGeenpickedup,andlaiddownalongthepathgivenbycFΫ.PNotethatifS :isconnected,thenUUST=c%isUUconnected.2W*e+"couldequallywellhaveusedthepieceofthelinkwhichwentunderthecrossingatv S,ofcourse.ThisSeifertdiagramisdenotedbyST=c^q,andde nedby^<$LűSLşwfeT (֍0c5L^Ƒۺ=<$KS^Kwfe N8 (֍0cr{:e (2:1)W*eFcallc^yamirr}'or-cutث.lWewillnotdealwithmirror-cutsagainuntilcorollary3.2,0wherefweprovethatconsideringbGothcutsandmirror-cutsgivesusnobGetterbGoundUUthanconsideringcutsalone.% ,Typb"esofCutW*e(~areinterestedincutswhichreducethesizeoftheSeifertdiagram.bW*estartbyde ningthis,)andothersubGclassesofcutswhichareeasiertohandle._[Bytheendofthechapter,wewillhaveextendedtheresultstoalargeclassofreducingcuts. ReducingCutsAUUcutc foraSeifertdiagramS _iscalledr}'educing7ҫwhen\'αszp(ST=c)simpleclosedcurvemadefrome,yPf r6,yPandthetwopGortionsofSeifertcircuitbGetween1them.eInordertoleave1this(whichitmust),8itmustcrossf *w,8andmustdoUUsointhesamedirectionasitcrossedeV.ff̟ffffffff̎"Semi-Lob"calCutsIfUa ګisUaSeifertcircuitofaSeifertdiagramS ѩ,'thenaseparatesS^2CWinto2compGo-nents,eachHisomorphictoB^2&..TwopGointsinS^2շn8a.aresaidtobGesep}'aratedѫbya&!whenUUtheybGelongtodi erentcomponents.2Acutc foraSeifertdiagramS issemi-lo}'caljwhennocircuita ofSseparatesa-pGointinim(c)8\im UT(S)naQ fromthespecialvertexofc.IPNotethateveryloGcalcutUUissemi-loGcal.(9CutSequencesIn+examiningthee ectofcuts,3wearenaturallyledtothefollowingde nitions.2GivengaSeifertdiagramS0,l[acutse}'quenceoflengthn 3forgS0misgasequenceofUUtheformC=(c1|s;c2;:::;cnq~);{UU34{#>CutsswhereUUforall1in,r,ciRѫisacutforSi1]1andϱSid=Si1 =ciTL:e (2:2)W*eUUde neHϱS0|s=CY=Snq~:e (2:3)R8AnyadjectivewhichcanbGeappliedtoacutcanbeappliedtoacutsequencemeaningUUthatitappliestoallthecutsinthesequence.2IfUUh misUUadeformationofS^2WandOC=(c1|s;:::;cnq~)isUUacutsequence,wede nehCFtobGethesequenceofembGeddingsk(hc1|s;:::;hcnq~):2F*or!aSeifertdiagramS zu,wede nerGcs(S)!ԣtobGethelargestn ܫsuchthatthereisareducingcutsequenceoflengthn vnforS p>.W*ealsode nesrGcs(S)(p5tobGethelargestJn .suchJthatthereisasemi-loGcalreducingcutsequenceoflengthnforS Ğ.ClearlyUUsrGcs(S)rcs(S)Me.2IfUUS _isUUaSeifertdiagramforalink`,thenbytheorem1.3,p-bi(`)szp(S)8rGcs(S)szp(S)8srGcs(S):e (2:4)W*eUUthereforede ne{ub(S)=szp(S)8srGcs(S);e (2:5)andthisistheuppGerboundonthebraidindexwithwhichthisthesisiscon-cerned.2GivenUUalinkdiagramL #,wealsode neA#ub(L)=ub(S);e (2:6)whereUUS _isUUtheSeifertdiagramassoGciatedwithL #.2The remainderofthischapterisdevotedtoprovingresultswhichmakecalculationUUofsrGcs(muchUUeasier.{UU35{$䙍Cutss2W*erestrictourattentiontomonotonecutsuntilthelastsectioninthischapter.W*ecZde nemsrGcs(S)0tobGethelargestn Nsuchthatthereisamonotonesemi-loGcal[reducingcutsequenceoflengthn PforS x,andl2`mcs(S)(tobethelargestn'慫suchOthatthereisaloGcalmonotonecutsequenceoflengthn 8forS ~.ʹClearly*,byUUtheorem2.4, -l2`mcs(S)msrGcs(S)srGcs(S):e (2:7)"n2Ifte gistanedgeofaSeifertdiagramS *,wede neSm8e tobGetheSeifertdiagramUUS ,UUwithouttheedgeeV.JTheoremT2.7%SuppGoseUUS _isUUaSeifertdiagramande Randf areparalleledges.2ThenҥΠl2`mcs(S) l2`mcs(Sm8e)andΠl2`mcs(S) l2`mcs(Sm8ef):!VʍProQof What\wewillactuallyproveissomethingslightlystronger,^\sothattheinductionUUgoGesthroughmoreeasily*.%2SuppGose5Sþis5aSeifertdiagramandr ~andm!areintegers,30r5m5wIandm299.qSuppGoseUUthat qe1|s;:::;emIareUUparalleledgesinS .qLetjS0(ޫ=Sm8e1S:::germ:!Vʍ2ThenUUl2`mcs(S)lmcs(S^0aƫ)ZeX.%2Thiswillprovetheresultasstatedbylettingm=2%*andr5=1UUor25asrequired.{UU36{%쿍Cutss2Theb.proGofisbyinductiononn=l2`mcs(S);.T,eandthecasen=0!Eistrivial.SuppGose,therefore,thatwn1Dandw(c1|s;c2;:::;cnq~)DiswalocalmonotonereducingcutsequenceforS 5.Thennoei(for1im/d)canbGetheactiveedgeofc1шbytheorem]2.2,^andsoc1\isacutforS^0 .Clearlyc1isstillloGcalandmonotoneforS^0').u2Bytheorem2.6,-eitherc1!crossesnoneoftheei ,orelseitcrossesallofthem.ConsiderUUST=c1andUUS^0aƱ=c11.2If c1bcrosses noneoftheei W, thentheinductivehypGothesisappliestoST=c1andUUS^0aƱ=c11,UUsogl2`mcs(ST=c1|s)lmcs(S0aƱ=c1|s):U^2If{c1Dcrosses{alltheei ȫ, thenitdividesthemasshownin gure2.8,andnote2.5applies.Leteiigofromthecircuitaictothecircuitbi ,}andletS^00bGeST=c1minusallthenewedgesfromtheaiKtoc1 ].GNowtwoapplicationsoftheinductivehypGothesisUUshowthat vsl2`mcs(ST=c1|s)lmcs(S00)lmcs(S0aƱ=c1|s):2HenceUUineithercase,Ml2`mcs(S)=lmcs(ST=c1|s)8+1lmcs(S0aƱ=c1|s)8+1lmcs(S0aƫ):pbff̟ffffffff̎&qҍÀCompressibilitySuppGoseUUc isUUacutforaSeifertdiagramS .qSupposefurtherthat {::qI,UX!S^2isUUanarcsuchthat,a(i)< z(I)8\im UT(c)= (@8I)Hr.)D(ii)< z(I)8\im UT(S)=;.{UU37{&NCutss2Ifwehavesuchanarc,I z(@8I)dividesim 9(c)intotwoparts,cv‫andc^፴v ,Iwherecv*FIcontainstheactivevertex.\Now %issaidtobGeac}'ompressingY2arc:forc ~whenj''(iii)<c^፴vHintersectsUUS .\2Ifn L%isnacompressingarcforc_,twecancompressc 0along toobtainanewcutUUc= N,UUwhere+imG(c= z)=cv.[8 ;\andMԱc= ^respGectsMtheorientationofc alongcv a.oGIfcwasloGcalormonotone,OTthenc= 14isUUstillloGcalormonotonerespectively*."@ IncompressibleCutsA!cut2issaidtobGec}'ompressibleҍwhen2ithasacompressingarc,andinc}'ompress-ibleotherwise.2F*orraSeifertdiagramS ƫ,9wede neil2`mcs(S).tobGethelargestn ~suchthatthereAisanincompressibleloGcalmonotonecutsequenceoflengthn /forS L.2lClearlyweUUcanextendequation(2.7)to`kmil2`mcs(S)lmcs(S)msrGcs(S)srGcs(S):e (2:8) ʼ2W*eaimtoshowthatalltheseinequalitiesareinfactequalities,1andwestartwithUUthe rsttwo.'dWٚallsLetUUSZ=(A;E;iA;iEm;[٫)]bbGeUUaSeifertdiagram.qAwall]ޫforS _isanembedding`!͙:qI,UX!S^2suchUUthat,a(i)<imG9(![٫)8\im UT(iEm)=;0.)D(ii)<9a2Aq:im4(![٫)8\im UT(iA)=!(@8I)iA(a8S^1 C) :.{UU38{'Cutss2TheUUSeifertcircuitiA(a8S^1 C)3iscalledtheend]ޫcircuitof! ꩫ.|2Intuitively*,;aXswallisanarcfromaSeifertcircuitbacktoitself,;avoidingeverythingUUelse.ɺ2Ifoc 3٫isoaloGcalmonotonecutforaSeifertcircuitS %S,thentherearetwooexternalwallsTinST=c2K,marked ūand ARin gure2.9,whichisamoredetailedversionof gure2.1.MThefactthattheseareinfactwallsfollowsfromthefactthatc isloGcalDandmonotone,andfrom gure2.8.TherearealsotwoDinternalͫwallsinST=c-,UUmarked ƫand fFin gure2.9.;j ff[eҍYffoff[0 퍒Q\StandardW*alls")ff[[Yff2|FigureT2.9X-ff[$iRob"omsGivenUUaSeifertdiagramS ,ar}'oomofUUS _isUUanembGedding],B rݫ:qS^1C,UX!S^2suchUUthatimq(rG)XPisoneof,a(i)<AUUSeifertcircuit.)D(ii)<AnevennumbGerofarcsorpointsinthecyclicorder(!1|s;s1;:::;!nq~;sn)<suchUUthat7(ii-i)PEachUU!i8qisUUtheimageofawall.4(ii-ii)PEachUUsiisUUa(connected,propGer)subsetofaSeifertcircuit.{UU39{(Cutss2A^roGom^satisfyingcase(ii)issaidtobeoflength#\n #,abasde nedinthatcaseabGove.gA7room7satisfyingcase(i)issaidtobeoflengthD0.gF*oraroomoflengthgreater`than0,bpart(ii)ofthede nitionofawallimpliesthatthesi mustallbGeWsubsetsofthesameendcircuit.vThiscircuitiscalledtheb}'ounding9circuitofthe +roGom.XdF*oraroomoflength0,gtheboundingcircuitoftheroomistheimageofUUtheroGomitself.ҍ2Iftgc insertedanexternalwall,sor GisaroGomoflengthatleast1inST=c.jQff̟ffffffff̎1LemmaT2.9SuppGoseUUc isUUalocalmonotonecutforaSeifertdiagramS .2ThenUUc isUUaroGominST=cL.ProQof Sinceڱc -isloGcalandmonotone,note2.5applies.8VThe(length2)roomnow7consistsofasubsetofaSeifertcircuitformostofc,=twointernalwalls(see gureUU2.9),andthepGointontheSeifertcircuitbetweenUUthem.Dff̟ffffffff̎{UU40{) Cutss2F*orUUthenextlemma,weusethenotationS0C=S%Sandt廱Sid=Si1 =ci TM(8i1)e (2:9)asUUintroGducedinequation(2.2),andonemorede nition.S2Ifc andc^0 ~:aretwocutsforaSeifertdiagramS cg,φwesaythattheyareambientisotopic inUUS _whenUUthereisadeformationh mofS^2Wsuchthat,a(i)<hj:imX:(Sa)y=id o:imǟ:(Sa).)D(ii)<hc=c^0ZЫ.2IfҦ(c1|s;:::;cnq~)8:andҦ(c^0l1|s;:::;c^0፴nq~)areҦtwocutsequencesforaSeifertdiagramS$,zwes=saythattheyareambientkisotopicwheneachciisambientisotopictoc^0;ZiinUUSi1ܫ.2IfC *i=(c1|s;:::;cnq~)JMisacutsequence,?wesaythatC aiswe}'aklyjdisjoint|whenforUUall1ix,,a(i)<imG9(ciTL)8\im UT(cj6)=;).)D(ii)<imG9(cj6)Z doGesUUnotintersecteitherofthetwostandardroGomsofci |.2Notethatthede nitionofweaklydisjointassumesthatthetransformationshownUUin gure2.1iscompletelyspGeci ed,notjustspeci eduptoisotopy*.LemmaT2.10SuppGoseCS ;\isCaconnectedSeifertdiagram,FR5Aisa nitesetofroomsofS X,FandCisUUanincompressibleloGcalmonotonecutsequenceoflengthn DforS .2ThenthereisaweaklydisjointincompressibleloGcalmonotonecutsequencet.(c0፮1|s;:::;c0፴nq~);ambientUUisotopictoC .,suchthatqYim|v[(c0፴iTL)8\im UT(rG)=; (81in;r52Rǫ):{UU41{*ōCutssProQof LetC7=(c1|s;:::;cnq~):TheproGofworksbyinductiononn p,andistrivialifn=0.WISuppGose,therefore,thatn1Sy.TW*earerequiredto nd(c^0l1|s;:::;c^0፴nq~)6andcorrespGondingdeformationsh1|s;:::;hnMasUUabGove.2Firstly*,Z$weY.useasmalldeformationh uSofS^2 tomovec1slightlyintogeneralpGositionUUwithrespecttoR .qF*oreaseofnotation,letc=hc1#ë,andwrite*/imF(Rǫ)=([ 7r72RSim(rG):'-NowUUwehavetoshowthatwecanisotopc o R ,andwedosobyinductionon\fm=jim (c)8\im UT(Rǫ)j:W*emayassumethatm>0,$soc ѫintersectsr52Rpz.Inparticular,$r i3mustbGeoflengthUUgreaterthan0sincec isloGcal.2LetUUthearcsofr tbGe/(!1|s;s1;:::;!nq~;sn):Sincevձc AisvloGcal,5itcannotintersectanyofthesi {!,5sowithoutlossofgeneralitycintersects7y!1 g.gLet7ythebGoundingcircuitofr 9bea.gWhenc ͫcrosses!1 g,=ritentersasimpleNclosedcurveconsistingof!1andapartofa~.Itmustleavebycrossing!1,'߫again,UUsincec isloGcal.qThereforeccrosses!1`atleasttwice.2Theؤsituationisnowasshownin gure2.10,xwheretheactivevertexofcis_notonthearcmarkedasc^፴v &.Ifc xintersects!1ڬalongthearcmarkedas |,itfmustdosoatleasttwice,jandsowecanusetheseintersectionsinstead,jandcontinueUUuntilc doGesnotintersect r.2Nowc2we ndthat 5satis esalltheconditionsofacompressingarcexceptfor(iii),oCbutjc (isjincompressible,soc^፴vOcannotintersectS g.Now!1cannotintersectS(either,DWand@S 5is@connected,sotheinterioroftheareaD«enclosedby Kandc^፴v{UU42{+! CutssLff[5+㞄xZff zff늟??j l\CutW*all"8ff늎늄xZffR}FigureT2.10X-ff[׍isfreefromSandc9x."Nowthereisadeformationh^'ofS^2whichissuppGortedonZasmallregularneighbGourhoodZofDBwhichleavesS ix xedandmovesc o !1nearUU r.]2ThisUUreduces?5jim (c)8\im UT(rG)jߍbyatleast2,soitreducesmGbyatleast2,andsobyinductionwecan ndadeformationh^ofS^2D whichmovesh^co R H>.GW*ecannowde neh1C=h^ȱh^handUUc^0l1C=h1|sc1,.荑2NowUUconsiderST=c^0l1L.qW*eseethatR,(h1|sc2;:::;h1cnq~)isUUaloGcalmonotonecutsequenceforST=c^0l1since?h1|sj:imX:(Sa)y=id o:imǟ:(Sa)б;and?isincompressiblesinceanycompressingarc ׫forh1|scigKwouldpullbacktoacompressingUUarch䍺11 t 2forci |.2Now!RMѫis!a nitesetofroGomsinST=c^0l1bylemma2.8,andc^0l1risaroGominST=c^0l15i[byUUlemma2.9.qLetr^G0l1andr^G0l2bGethetwoUUstandardroomsinST=c^0l1L.qNowletSRǟ0=RL[8fc0፮1|s;rG0፮1;rG0፮2g:{UU43{,,Cutss2ByinductionthereisaweaklydisjointincompressibleloGcalmonotonecutsequence=(c^0l2|s;:::;c^0፴nq~)6and=deformationsh^0l2|s;:::;h^0፴n12suchthatforall2in.andr52Rǟ^09T,N,a(i)<h^0;ZiTLj:imX:(S O \cmmi5i0ncmsy51 Ǯ)#BL=id o:imǟ:(Si1 Ǯ) .)D(ii)<h^0;ZiTLh1|scid=c^0;Zil.''(iii)<imG9(c^0;ZiTL)8\im UT(rG)=;Ƕ.2F*orUUall2in,r,de nehid=h^0;ZiTLh10masrequired.2Now c^0l1dcannot intersectanyc^0;Zi=with2in.*6,zbGecausec^0l1C2Rǟ^0$. Simi-larly*,űc^0;Zi6hcannot|intersecteitherstandardwallofc^0l1 ʫ,bGecauser^0l1|s;r^0l2C2Rǟ^00ݹ.;Hence(c^0l1|s;:::;c^0፴nq~)RisUUweaklydisjointasrequired.ff̟ffffffff̎, 8|Sho9wingTthat8DF cmmib10ilmcs=msrcsNThe~remainderofthissectionisdevotedtoaproGofthatil2`mcs(S)=msrcs(S)a.TheUproGofisbyinductiononmsrcs(S)-Lm,usingtheinductivehypGothesisH n,forn2N9V:#H+n4U:<SuppGoseUUS _isUUaconnectedSeifertdiagram,mnr,andmsrcs(S)mC4r.<ThenUUil2`mcs(S)m@Z.2NoteUUthatH 0ʫistriviallytrue.2TheUUfollowinglemmaisapartialconversetotheorem2.7.LemmaT2.11SuppGoseUUH nQ.2SuppGose)S 'is)aconnectedSeifertdiagramwiththreemutuallyparalleledgese"ĝ,UUf andUUgv8.qSuppGoseUUmn"IandٱmsrGcs(Sm8e)m:{UU44{-4-Cutss2ThenHmsrGcs(S)m:UoProQof The proGofisbyinductiononm ,andistrivialifm=0$,sosuppGosethatm199.qЍ2ByH ?n0,let(c1|s;:::;cm):&bGeanincompressiblelocalmonotonecutsequencefor,Sm8e«.d0Consider,e just,asanarcembGeddedinSm8e.d0Leth bGeadeformationofS^2 ,j xedonSm8el",whichmovesc1|,intogeneralpGositionwithrespecttoe}.GPF*oreaseUUofnotation,letc=hc1#ë.qBytheorem2.6,oneofthefollowingholds.,a(i)<cC̫crossesUUneitherf norgv8.)D(ii)<cC̫crossesUUbGothf andgv8.2In]case(i),theproGofoftheorem2.6showsthatc doesnotcrosse ի,socislnaloGcalmonotonecutforS !«,4(Sm8e)=c=ST=ce[,4andtheresultfollowsbyinduction.2InBcase(ii),(ytheorem2.2impliesthatneitherf 3jnorg [gcanbGetheactiveedgeofLcY'.NowLc ^scrossesLf~andLisloGcalandsimple,1Isoinordertoleavethesimpleclosedcurveconsistingoff ɫ,$GeH,$GandthetwosectionsofSeifertcircuitbGetweenthem,c kmustacrosse ëanoGddnumbGeroftimes.WSupposethatc Bcrossese ëmorethan once.DThenwehavethesituationshownin gure2.11,wheretheactivevertexUUofc isnotonthearcmarkedasc^፴v i~.2If ±c _crosses ±e 酫in thearcmarked ߫,+Fthenitmustdosoatleasttwice,+FandwecannFusethispairofcrossingsinstead,tandrepGeatuntilc 0gdoesnotcross c.Now '-satis esrtalltheconditionsofacompressingarcforc 8ëexceptfor(iii).#Butcisincompressible, so۱c^፴v߫cannotintersectSm8e.RNow ^ӫandc^፴v߫formasimpleclosedcurve#whichdoGesnotintersectSm8e,-andSm8e isconnected,sinceS ,isande ٫isparalleltof,,sotheareamarkedasD ֫in gure2.11isfreefromSm8eHԫ.;Thereforethere^isadeformationofS^2isuppGortedonasmallregularneighbourhoodofDwhichUUmovesc o e Rnear r.qThisreduces&P>jim (c)8\im UT(e)j{UU45{.>\CutssLff[5/rxZff zffΐ??]gD\DoubleCross"8ffΐΐxZffR}FigureT2.11X-ff[<byUUatleast2,andmustthereforeterminate.W2AttheendofthisproGcess,wehaveacutc^0 $,ambientisotopictoc8,whichintersectsНe I;exactlyНonce.Leth^02jbGethedeformationofS^2justconstructed,nsothatUUh^09c=c^0$«. 卑2Consideringc^0/asacutforS < ,itisclearlyloGcalandsimple,sotheorem2.6ensuresUUthatc^0 ̾ismonotone,soitisreducingbytheorem2.4.2Nowf(h09c1|s;:::;h0cnq~)ʍisUUamonotonesemi-loGcalreducingcutsequenceforSm8e눫,andthereforeбmsrGcs((Sm8e)=c09)m1:ConsiderationPof gure2.8showsthatwecanapplytheinductivehypGothesisthreeUUtimestoremoveUUeachedgeshownthereinturn,andthusshowthat"msrGcs(ST=c09)m81;andUUhenceױmsrGcs(S)mʍasUUrequired.ff̟ffffffff̎{UU46{/J;CutssLeaves'F*orUUthemainstepintheproGof,weneedanotherde nition.qSuppose-/a;cq:S^1C,UX!S^2areUUsimpleclosedcurveswhichintersectingeneralpGosition, im*V(a)8\im UT(c)6=;;andWv"2im (c)8nim UT(a):NThen a qpdivides S^22into twocompGonentsisomorphictoB^2,!*oneofwhichcontainsv#QY.dW*e4de neBq^0TګtobGethecomponentcontainingv ,andBtobGetheothercompGonent.8 Consider!thegraphT 㰫,whoseverticesaretheconnectedcompGonentsofmBQn8im UT(c),-,sjoinedmbyanedgewhenthecompGonentsareadjacent.ThisTisatree,sinceeveryedgeseparates.xAcompGonentofBQn8im UT(c)0g1whichcorrespGondstoaleafofthistree,%thatisanon-separatingvertexofT U,iscalledale}'af.]LetLbGe3aleaf.fThenthesituationisoneofthoseshownin gure2.12(i)-(iv).A3leafof+mtypGe(i)or(ii)issaidtobeago}'od3leaf,3and+maleafoftype(iii)or(iv)isab}'adleaf.DTff[|܍pff~ff[;~R\GoGodBad"4ff[[pffR}FigureT2.12X-ff[{UU47{0P CutssLemmaT2.12.LetUUa,UUc andUUv gbGeUUasabove.2ThenUUeitherthereisagoGodUUleaf,orelsethereareatleasttwoUUbadleaves.ProQof Since im*V(a)8\im UT(c)6=;;]theBtreehasanedge,FhenceithasatleasttwoBleaves.kEitheroneoftheseleavesisUUgoGod,orelseallarebad.ff̟ffffffff̎!AnOrderonCuts.W*eLnowneedtosetupanorderingoncutssothattheinductioninthefollowingproGofJcangothough.n,SupposeSZ=(A;E;iA;iEm;[٫)]isaSeifertdiagramandc isaUUcutforS .2W*eUUde nethegr}'ading7ҫofc tobGethepairofintegersQng[٫(c)=(jim (c)8\im UT(iA)j;jim(c)8\im UT(iEm)j):` (2:10)2IfUUc andUUc^0 ̾areUUcutsforS ,let{g[٫(c)Éo=(s;t)andBg[٫(c09)Éo=(s09;t0):QThenweorderthecutslexicographicallybyc0S!.)D(ii)<cC̫isUUloGcalandcompressible.''(iii)<cC̫isUUloGcalandincompressible.2W*eUUwilltakethesethreecasesseparately.$hʍCase J(i) T*akecsomea2A ӫsuchthatim(c)8\iA(aS^1 C)6=;eu,andletv ;bGetheactiveovertexofcdJ.ZNowlemma2.12appliestoaY,7c andvE,,7sooneofthefollowingmustUUhold.&q(i-i)<ThereUUaretwoUUbadleaves,UUL1WandUUL2.#(i-ii)<ThereUUisagoGodUUleaf,L #.E܍2In-case(i-i),#wehaveapre-diagram[ST=c]ofsizeszp(S)81.C.NNowifwechange[ST=c]H[atthefourobviouscrossingsonthebGoundariesofL1׫andL2toanequivqalentUUpre-diagramQ|asshownin gure2.13,thenbylemma1.4,4eq/Sszp(ST=c)sz(Q)=sz([ST=c])8+1=sz(S);whichUUcontradictsthefactthatc isreducing.E܍2In{)case(i-ii),ĝc issemi-loGcal,ĝandsocannotcrossanycircuitoredgeontheothersideofa g`fromvի.Thereforethesituationisshowninthe rstpartof gure2.14,orthesame gurewiththeorientationsreversed.8Withoutlossof{UU49{2^Cutssi`ff[o荍cffprff[5 xX\TwoBad".ff[[cffR}FigureT2.13X-ff[̍f[ff[fɐffN։ff[[Fѕ\GoGodLeaf"ULff[[ffR}FigureT2.14X-ff[generalitywemayassumetheformer.TheareaGLrepresentsthoseedgesofSwhich~areincidentontheoutsideofthepieceofa ,onthebGoundaryofL z.wCTheareaUUF|rrepresentsUUallofS _inL #.獑2LetUUc^0 ̾bGeUUthecutshowninthesecondpartof gure2.14andletҏBg[٫(c09)=(s0;t0):{UU50{3fCutssThends^0Q=s82-O,.sodc^0Q<c^.1"Figured2.15showsthesituationsinST=candST=c^0ܔ.Theareahmarkedas2GinST=c^0nconsistsofedges,AtwiceasnumerousasG 4,Aformedfrom+thechangeshownin gure2.8.cThisshowsthatc^0 yismonotone,4semi-loGcalandUUreducingforS .0Jff[T`ff%cff[p.\GoGodLeafCut"j͉ff[[`ffR}FigureT2.15X-ff[2BybassumptionmsrGcs(ST=c)mM=",%andST=cisconnected,soH #n,saysthatthere isanincompressibleloGcalmonotonecutsequence(c1|s;:::;cm)8forST=cp.QLetr%EbGetheroomoflength1inL dwhichincludesthewall! x~asshownin gure2.15.De ne*ktheinsideƫofr tobGetheconnectedcomponentofS^2mn8im UT(r)1~containedinL$֫.v2Bylemma2.10,wemayassumethat(c1|s;:::;cm)8Kisdisjointfromrܫ.QW*ecansplitUUthissequenceintotwodisjointsubsequencesq(c0፮1|s;:::;c0፴lȫ){UU51{4jSCutssand|#\(c0፴l `+1 1˱;:::;c0፴m)|whereUUthelattersubsequenceconsistsofthosecutsciRѫinsideri.2Then(c0፮1|s;:::;c0፴lȱ;c0፴l `+1 1˱;:::;c0፴m)isUUclearlyaloGcalmonotonecutsequenceforST=cL.|2ConsiderS<˫andS^0asshownin gure2.16.@[Letp bGethenumbGerofedgesinE&.The`onlydi erencebGetween`Szѫand`S^0,bup`toambient`isotopy*,is`thattherearey2edgesbGetweenythecircuitsb18andb2inSs$,Band2pxedgesbGetweenyb1andb2*9inUUS^0.J ˉff[ ffFff[wA\GoGodLeafPrune"pff[[ ffR}FigureT2.16X-ff[{UU52{5pCutss2ClearlyqJ(c^0l1|s;:::;c^0vlȫ)5|isqJaloGcalmonotonecutsequenceforS+,xHandsobythe-oremUU2.4,S[msrGcs(S)l2`:V2IfUUp=0thenUUlxmn/|;UUsobyH nQ,theorem2.7andequation(2.8),Za lxil2`mcs(S)lmcs(S)lmcs(S0)msrGcs(S0):t92IfUUp1,UUthenlemma2.11canbGeused2p82!timestoshowthatmsrGcs(S0)msrcs(S)l2`:2Therefore,UUwhateverthevqalueofp]U,S[msrGcs(S0)l2`:NowUUbyH nQ,thereisanincompressibleloGcalmonotonecutsequenceFCݟ00g=(c00፮1r;:::;c00፴l)for͛S^0 F.DNow͛r eJis͛aroGominST=c^0˫,andhenceitisaroominS^0 F,sincetheboundingcircuith4isinS^0"߫.cThereforebylemma2.10,wemayassumethatC Au00doGesnotintersectUUri.r92NowUUtheincompressibleloGcalmonotonecutsequence#\(c0፴l `+1 1˱;:::;c0፴m)forST=(c;c0፮1|s;:::;c0፴lȫ)rinsideUUr tisUUalsoacutsequenceforST=(c09;c00፮1r;:::;c00፴l):ThisUUisstillloGcalandmonotone,thereforereducingbytheorem2.4.2Thus](c00፮1r;:::;c00፴l;c0፴l `+1 1˱;:::;c0፴m)risWamonotonesemi-loGcalreducingcutsequenceforST=c^0 ,soc^0«satis esP u,con-tradictingUUtheminimalityofc0.{UU53{6uhCutssCaseT(ii) LetUU ǫbGeUUacompressingarcforc0.qLetʁgϱg[٫(c= z)=(s09;t0):Thens=0andt^0Q<tT, )soc= В<c%c.XThesituationisnowshownin gure2.17.TheareasF andG representtwopartsofS TB,jtheareaE 30bGetweenthemrepresentspGossible /edgesjoiningthem,9andthesetogethercompriseallofS .VTheactivevertexUUofc isinF '.Íhyff[dfHrffLffʕg[S=#\CompressBefore"WDffʕgʕgrffR}FigureT2.17X-ff[ H,2InywST=cn,theywsituationisstillasshownin gure2.17,exceptthatnowc Fɫisapart#ofaSeifertcircuit,, Visawall,,andtheedgesinEhavebGeendividedasshownUUin gure2.8.qNowST=c%isconnected,andʁ&wmsrGcs(ST=c)m;soUUH nիsaysUUthatthereisanincompressibleloGcalmonotonecutsequenceC=(c1|s;:::;cm){UU54{7~ԍCutssforST=c.[9Nowc^፴v7and stogethermakearoGomr joflengthatleast1inST=c,4sobylemmaUU2.10wemayassumeC isdisjointfromri.n02FigureH2.18showsthesituationinST=(c= z)%{.]Thedottedline ]issimplythelineUUlabGelledc^፴vӫin gure2.17whichnoneofc1|s;:::;cm1Gcross.Yhyff[dfHrffLffʕg[S?Z\CompressAfter"WDffʕgʕgrffR}FigureT2.18X-ff[(2LetXE nƫbGeXthesetofedgesinS whichc^፴vV٫crosses.USupposee2EK.UThenc ismonotone,Phencesimplebytheorem2.3,andsowehaveoneofthetwosituationsshownԋin gure2.19.FLetheXbGeadeformationofS^2ësupportedonasmallregularneighbGourhood ^ofe whichmovesc lneare intothecorrespGondingpositionshowninc gure2.20.UDe nethedeformationh ŽofS^2 stobGethecompositeoftheheoverallUUe2EbH.2Clearly.hCXis.amonotonesemi-loGcalreducingcutsequenceforST=c,dsinceall?.thath ASdoGesistomove?.someedgeintersectionsontotheneighbGouringSeifertcircuit.W*enowaimtoprovethathCvisalsoamonotonesemi-loGcalreducing{UU55{8<CutssJ܉ff[5ku9ff,ffS=VZ\BubbleOne"7#ffSSu9ffR}FigureT2.19k+v\u9ff,ff=Vc<\BubbleTwo"7#ffu9ffR}FigureT2.20X-ff[cut{SsequenceforST=(c= z)&,.LetX^PbGetheareainS givenbytheunion,overalle2E5),*oftheareasXep-shownintheappropriatehalfof gure2.21,andletXbGetheunionofX^̚withasmallregularneighbourhoodoftheroomr .ZNowST=candCST=(c= z)(areCidenticaloutsideX  ,see gure2.8,andnoneofthehciQintersectX'.b.qThereforeUUhCFisUUamonotonesemi-loGcalcutsequenceforST=(c= z)&.4j2T*oUUshowthathCFisreducingforST=(c= z)&,let=HSm _=ST=c=hCandJIS m _=ST=(c= z)=hCA:{UU56{93Cutss&i?ff[ݍU܄Ѳaff.ffk뮍lf\BubbleThree"e`MffѲaffR}FigureT2.21X-ff[㸍ThenSmandS^ mdi eronlyintheareaX (.Moreover,G isapieceofSeifertcircuitinS^ m4withnoincidentedges.RSuppGosewemovethispieceofcircuitfrom 'toPJ f.pThisPJgivesusapre-diagramPoandanassoGciatedSeifertdiagramwhichwe,vwillcallS^ mث.*ThenS^ mNandSmdi er(uptoambient,visotopy)onlyinthesignsUUoftheedges,andthereforehaveUUthesamesize.'㍑2NowUUlemma1.4saysthat3{4szp(Pc)sz(S m):Hencez-szp(ST=(c= z)=hCA)=sz(S m)=sz(Pc)W^szp(S m)=sz(Sm)=sz(ST=c)8m=sz(ST=(c= z))8m;and4nowtheorem2.1impliesthathCisreducingforST=(c= z)%g.UThusc= asatis esP%㺫,UUbutc= В<c$w,whichcontradictstheminimalityofc0.{UU57{:CutssCasej(iii) Inthiscase,!ST=c?isconnectedandmsrGcs(ST=c)mMZ,!andsoH @nimpliesUUthatil2`mcs(ST=c)mI /,so Zil2`mcs(S)m8+1asUUrequired.ff̟ffffffff̎)1H2NowUUH 0ʫandUUlemma2.13showthatH nիholdsforalln U.!v0TheoremT2.14SuppGoseUUS _isUUaconnectedSeifertdiagram.2ThenEil2`mcs(S)=lmcs(S)=msrGcs(S):#SProQof ByHH n s,=ıil2`mcs(S)msrGcs(S)b=\.NowHequation(2.8)givestherequiredresult.7dff̟ffffffff̎0`)RemovingMonotonicityTheoremUU2.14improvesUUequation(2.8),forS _connected,to kmil2`mcs(S)=lmcs(S)=msrGcs(S)srGcs(S):` (2:11)TheUUaimofthissectionistoshowthatthelastqȫissharpaswell.)1HCo-lob"calCutsA5cutEc eforEsSeifertdiagramS ޫisc}'o-local ΫwhenEitonlycrossesoneedgeofS ͙,${itsactiveTedge,ؔanditonlycrossesthatedgeonlyonce.A9co-loGcalcutistriviallymonotone.{UU58{;퍟CutssTheoremT2.15GSuppGoseUUS _isUUaconnectedSeifertdiagramwithsrcs(S)n7.2ThenUUmsrGcs(S)n@mU.ProQof The6proGofisbyinductiononnandistrivialwhenn=0W,,.sosupposen16r,UUandlet(c1|s;:::;cnq~)7@HbGeasemi-localreducingcutsequenceforS .2LetIe0QbGeItheactiveedgeofc1 旫.\Consideranedgee6=e0!NUofS ˝.Thenc1maycrosse oneithersideofthevertexofe(orbGoth)._WLetheJJbeadeformationofS^2suppGorted xonasmallregularneighbourhoodofe whichmovesthesecrossingsontoUUtheadjacentSeifertcircuitsasshownin gure2.22. bff[jƍdffR֔ffdŸYg.\PushBoth"SOffdŽd„ffR}FigureT2.22X-ff[-E2Similarly*,glet,he0{bGe,adeformationofS^2supportedonasmallregularneigh-bGourhoodofe0whichmovesallthecrossingsofc Yande0o e0 3,YexceptforthecrossingUUattheactivevertexofc1 %.{UU59{<Cutss2Leth :-bGethecompositeofallthehexoveralltheedgese 7ofS ,andletc=hc1" .NowKc TqisKco-loGcalandsemi-local,butST=c{andST=c1areambientKisotopicSeifertdiagrams,UUsoc isreducing.qThussrGcs(ST=c)=srcs(ST=c1|s)n81;soUUbyinduction햱msrGcs(ST=c)n81;soUUmsrGcs(S)nCªasUUrequired. ff̟ffffffff̎!2Finally*,mktherefore,theoremh2.15improveshequation(2.11),mkforS connected,tokmil2`mcs(S)=lmcs(S)=msrGcs(S)=srGcs(S):` (2:12)W*eznowshowhowthisequation(2.12)canbGeused.6Intuitively*,܃itmeansthatanysemi-loGcalreducingcutsequencemayaswellbGeincompressible,localandmonotone,andassuchitenjoysfarmorepropGerties,enablingustoproveresultsabGoutUUsrcs(whichUUmakecalculationconsiderablyeasier.&OrParallelEdgesTW*estartbyprovingsomeusefulresultsregardingparalleledgesinS ;0,andhowtheyUUa ectsrGcs(S)$O.TheoremT2.16SuppGoseUUS _isUUaconnectedSeifertdiagramande Randf areparalleledges.2ThenJ}srGcs(S) srGcs(Sm8e)andJ}srGcs(S) srGcs(Sm8ef):MProQof UseUUtheorem2.7andequation(2.12).ff̟ffffffff̎{UU60{=CutssTheoremT2.17 )SuppGose#S ˚is#aconnectedSeifertdiagramwiththreemutuallyparalleledgese3$,f'jandUUgv8.2Then c_srGcs(S)=srcs(Sm8e):捑ProQof SinceUUH nիholdsUUforalln U,lemma2.11saysthat c_srGcs(S)srcs(Sm8e):TheoremUU2.16saysthatc_srGcs(S)srcs(Sm8e): pbff̟ffffffff̎$z2ThisresultmeansthatweneedneverconsiderSeifertdiagramswithmorethan7Mtwomutuallyparalleledges.gThenextresultsaysthatifaSeifertdiagramdoGesUUhavetwoparalleledges,thenwithoutlossofgeneralitytheyareadjacent.R2Givennanedgee ݫinaSeifertdiagramS «,twede neSm+8eqtobGetheSeifertdiagramUUS _withUUtheedgee Rreplacedbytwocloseparalleledges.CorollaryT2.18 )SuppGoseUUS _isUUaSeifertdiagramande Randf areparalleledgesinS .2Then J}srGcs(S)=srcs(Sm+8ef):捑ProQof ByUUtheorem2.17, q6srGcs(S)=srcs(Sm+8e)=srcs(Sm+8ef):pbff̟ffffffff̎{UU61{>덟CutssSpb"ecialSeifertDiagramsgPSuppGose(a eis(aSeifertcircuitinaSeifertdiagramS C|.AThenaisasimpleclosedcurve,UUsoitdividesS^2Wintotwopieces.ΠDe nitionT2.19W*esaythata isoftyp}'e=IIwhenbGoththesepiecescontainothercircuitsofS 6,and a \Pis oftyp}'eI botherwise.xIneithercase,6letthetwo piecesbGeB1%[andB2˫,isomorphicCtoB^2չ.kThisde nesSeifertdiagramsS1&andS2 ),GjwhereSiconsistsofthecircuitsandedgesofS ܫwhichlieinBiث.WNotethata isacircuitinbGothS1andPS2 .p:NotePalsothatifa ꌫisoftypGeI,oneoftheSiohasjustthecircuitaޫ,Qandhence%noedges,/RsinceanedgemustgobGetweentwodi erentcircuits.aTheotherSi*isUUallofS .ΠTheoremT2.20SuppGoseUUa isUUaSeifertcircuitinaSeifertdiagramS .2ThenQIsrGcs(S)=srcs(S1|s)8+srcs(S2|s):5ProQof SuppGoseűsrGcs(SiTL)=ni TM(i=1;UP2):ΡThenubyequation(2.12),2il2`mcs(SiTL)=niE",2sothereisanincompressibleloGcalmonotoneUUcutsequence(ci፮1|s;:::;ciniv)forTWSi j.qrByTWlemma2.10,TweamyassumethatthiscutsequencedoGesnotcrossa,andUUisthereforestillaloGcalmonotonecutsequenceforS .qThereforePꍒ T(c11|s;:::;c1፴n1 X;c21;:::;c2፴n2 X)isUUaloGcalmonotonecutsequenceforS ,soyl2`mcs(S)n1S+8n2|s:{UU62{?7Cutss2Conversely*,UUsuppGosethatPsrGcs(S)=n:ThenUUbyequation(2.12),Yʱil2`mcs(S)=n;so9thereisanincompressibleloGcalmonotonecutsequenceC Coflengthn k forS j.Now0a 琫is0aroGomoflength0inS ,sobylemma2.10,wemayassumethatnoneofthecutsinC Mcrossa:6.PXNowwecanpartitionC Mintotwocutsequences,CW1KЫandCG2 F,whereCC iisCaloGcalmonotonecutsequenceoflengthni>forSi ,G4andn1S+8n2C=n7̌.ThereforelsrGcs(S)l2`mcs(S1|s)8+lmcs(S2|s);andUUequation(2.12)givestheresultasrequired.ff̟ffffffff̎$2ThisresultmeansthatwemayaswellrestrictourattentiontoconnectedSeifertdiagramswithnocircuitsoftypGeII.Thesediagramsarecalledsp}'ecial,followingUU[27].{UU63{@Gs!}Chapter3.fgGraphs!7De nitionsW*ewillbGeusingasmallamountofgraphtheoryinwhatfollows,sowecollectto-getherherethede nitionswewillneed.ALSee[6],forexample,foranintroGductiontoUUgraphtheory*.2F*romnowoninthisthesis,agr}'aphK$isatripleG=(V9;E;f)=K,whereVandE)gEareUU nite(disjoint)sets,and |xfO:qEZ!ffv[;wDgjv;w 2V9;v"6=wDg:e (3:1)Inwmotherwords,paralleledgesareallowed,loGopsarenotallowed,andedgesareunoriented.vIftherearenoparalleledges,Hthatisiff isinjective,HwesayGissimple[.2W*e0saythatVoisthesetofvertic}'es!,8andEVisthesetofedgesիasusual.eF*ore2E6ͫ, theπmembGersoff(e)aretheend verticesofew.IThetwoendverticesofanUUedgeareadjac}'entث.!2AUUplanaremb}'edding7ҫforGvisapairi=(iVɱ;iEm)8Asuchthat,a(i)<iV c:qV,UX!S^2t.)D(ii)<iE 3-:qEm8I!S^2^.''(iii)<iEm(e8@8I)=iVɫ(f(e))Y.'GC(iv)<iEmj:Eb}(In@n9I)nAZisUUinjective.2This=de nestheusualconceptofaplanarembGedding.?~AgraphisplanarwhenithasaplanarembGedding.Aqplanegraphisapair(G;i)G,Jwherei graphis2-c}'onnected]ޫwhenUUitisconnectedandithasnoseparatingvertices.i2AUUgraphisbip}'artitewhenthereisapartitionoftheverticesJV=V1St8V2suchUUthateveryedgehasendverticesinbGothV1randV2 .*%x4TheSeifertGraph4A(spGecial9SeifertdiagramS ƫhasaconnectedplanegraphG(S)associatedwithit,Q8calledP0theSeifert,gr}'aphc,formedbycontractingeverycircuitinS Utoavertexin±G(S)I,^andthusmakingeveryedgeinSثanedgeinG(S)I.ASinceanedgeofS(runs-fromananti-cloGckwise-circuittoaclockwisecircuit, asseenfromthatedge,^G(S)0{isbipartite.JThisgraphhasbGeenknownsince1930[3],^andhasbeenstudiedUUrecently[28],[32].2GivenCaconnectedplanebipartitegraphG ,zwecanconstructaspGecialSeifertdiagramPS(G))byPexpandingeachvertexintoasmallSeifertcircuit.*Thecon-structionUUisobvious,exceptfortheorientation,sinceG(S)=G(S)E$=.qNowN΍(SGS(G)=Ge (3:2)andN΍OʱSG(S)=ST;e (3:3){UU65{B3aGraphsssoR connectedplanebipartitegraphsareinbijectivecorrespGondenceuptoambientisotopyUUwithsetsofspGecialSeifertdiagramsfST;Sg&j.qClearly~@srGcs(S)=srcs(S);e (3:4)sosinceweareinterestedinsrGcs(S))3yforspGecialS ,NYwemayaswellworkwithconnectedUUplanebipartitegraphs.uq2W*eUUthereforede neѬsrGcs(G)=srcs(S(G))e (3:5)㍑andKsub(G)=ub(S(G))e (3:6)for)aconnectedplanebipartitegraphG h.BBycorollary2.18,wemayassumethat=anyparalleledgesinGXareadjacent.iThereforewecantreatGXasasimplegraphxwitheachedgelabGelledn1toshowwhetheritrepresentsoneedgeorn'radjacentUUparalleledges.uq2AsimplegraphwithapGositiveintegerlabGeloneachedgeiscalledalab}'elledgraph.qF*ormally,UUalabGelledgraphisa4-tuple~(V9;E;fV;g[٫)suchUUthatuq,a(i)<(V9;E;f)dnaisUUasimplegraph.)D(ii)<g͙:qEZ!N n8f0gb.2AnedgelabGelled1iscalledsingle[.@Anedgelabelledmorethan1iscalledmultiple[."^By:theorem2.17,t>weneedonlyconsiderlabGelledgraphswithedgeslabGelledUU1or2,buttheextrageneralitymakesproGofssimpler.#ō2-ConnectednessSuppGoseUUa ګisUUacircuitinaspecialSeifertdiagramS .qThen֍G(S)8aT͍UU+3UU=ٴn 3 a tpi=1gG0፴iTL;e (3:7){UU66{CaGraphsswheren1andeachG^0;Zi&isnon-emptyandconnected.Notethatn>1ifandonly&ifa lisseparating.b5W*ede neGi~TtobGethesubgraphofG(S)zspannedbyG^0;Ziand'acW.^ W*e'thende neSiatobGethepartofS 颫correspondingtoGiK?.^ NotethatalltheUUSi areUUconnected.!!RTheoremT3.1SuppGoseUUa isUUacircuitinaspecialSeifertdiagramS .2ThenXe7srGcs(S)=>n X tմi=1㉱srcs(SiTL):+ፑProQof TheWproGofisbyinductiononn,$andistrivialifn=1 @,sosuppGosen2 @.2AsXusual,%a /dividesS^2]intotwopieces,%B andBq^0 ʫ.MsSinceS isspGecial,%Bq^0 ◫,%say*,isUUempty*.qSincea ګisseparating,Bisnotempty*.2Nowjim (S1|s)8na1andjim (Snq~)8na2arejconnectednon-emptysubsetsofS^2 uA.#Con-siderrthesimpleclosedcurve Āwhichseparatesthem,9andliesclosetoim (S1|s)8na./.Thisisdisjointfromim =(S)8na*ڠ,andmustcrossa,sinceS isconnected.,Thisgivesusٗasetofwalls \8B owithendcircuita "ǫ,andoneofthesewalls,!isay*,mustseparaterim`(S1|s)8na3"fromrim`(Snq~)8na4inrB %.ʎThersituationisnowshownin gure3.1.2Now kܫdivides im(S)8na0into twoparts,ӱT^c0l1%fandT^c0l2,.(F*orjY=1UUor2/s,letTjbGetheSeifertdiagramcorrespondingtoT^c0;Zjo[8av.yNowT^c0;ZjisadisjointunionofconnectedpcompGonentsofim(S)8a/d.UHence,byreorderingtheSibifnecessary*,weUUmayassumethattheremovqalofa ګseparatesT1into X0S1|s;:::;SkandUUT2intoXSk+B+1 ;:::;Snq~;{UU67{DaGraphssč,p/ff[\9&ffg䪉ffzNZxMS\SeparatingW*all"Iމffzzff2|FigureT3.1X-ff[k'whereUU1k.ItUUisthereforestillaloGcalmonotonecutsequenceforS .qInfact,8 T(c11|s;:::;c1፴n1 X;c21;:::;c2፴n2 X)isUUaloGcalmonotonecutsequenceforS ,soyl2`mcs(S)n1S+8n2|s:( 2Conversely*,UUsuppGosethatPsrGcs(S)=n:ThenUUbyequation(2.12),Yʱil2`mcs(S)=n;so9thereisanincompressibleloGcalmonotonecutsequenceC Coflengthn k forS j.Now#bylemma2.10,-wemayassumethatnoneofthecutsinC acrossr$.aNowwecan1partitionC j"aswell.LemmaT3.3SuppGosen(c1|s;:::;cnq~)5sisnaweaklydisjointloGcalmonotonecutsequenceforaSeifertdiagramUUS .2ThenUUthereisadeformationh mofS^2Wsuchthat8zΟ(hc1|s;:::;hcnq~)isUUastronglydisjointco-loGcalsemi-localreducingcutsequenceforS .ProQof LetC7=(c1|s;:::;cnq~):ፑW*e1"willshow rstthatforall1in,NM,8_ciintersectsavertexofS v,8_crossingtheedgeW=therefromlefttoright.wSuppGosethisisnotthecase,Wandleti nbeminimalsuchUUthatitfailsforci |.qUsingthenotationS0C=S%Sand8zSm _=Sm1=cm{UU71{H󅍟aGraphssas$inequation(2.2),letj bGeminimalsuchthattheactivevertexofcioisinSj ݗ.SinceM#ciBmisM#acutforSi1,NDZjY<iͫ.o ByassumptionjY>0S6,Nsotheactivevertexofciis Cformedbythecutcj ʫ.YButcj isloGcalandmonoto