Damian Heard

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<p>Computationofhyperbolicstructureson3-dimensional orbifoldsDamianHeardSubmittedintotal fullmentoftherequirementsofthedegreeofDoctorofPhilosophyDecember2005DepartmentofMathematicsandStatisticsTheUniversityofMelbourneAbstractThecomputer programs SnapPeabyWeeks andGeobyCassonhaveproventobepowerfultoolsinthestudyofhyperbolic3-manifolds. Manifoldsarespecialexamplesof spacescalledorbifolds, whicharemodelledlocallyonRnmodulo-nitegroupsofsymmetries. SnapPeacanalsobeusedtostudyorbifoldsbutitisrestrictedtothosewhosesingularsetisalink.Onegoalofthisthesisistolaydownthetheoryforacomputerprogramthatcanworkonamuchlargerclassof3-orbifolds. TheworkofCassonisgeneralizedandimplementedinacomputer programOrbwhichshouldprovidenewinsightintohyperbolic3-orbifolds.Theothermainfocusof thisworkisthestudyof 2-handleadditions. Givenacompact3-manifoldMandanessential simpleclosedcurveonM, thenwedeneM[] to be the manifold obtained by gluinga 2-handle to Malong . If lies on a torus boundary component, we cap o the spherical boundary componentcreatedandtheresultisjustDehnlling.Thecasewhenlies onaboundarysurfaceof genus 2is examinedandconditions on guaranteeing that M[] is hyperbolic are found. This uses a lemmaof ScharlemannandWu, anargument of Lackenby, andatheoremof MarshallandMartinonthedensityof strippackings. Amethodforperforming2-handleadditionsisthendescribedandemployedtostudytwoexamplesindetail.Thisthesisconcludesbyillustratingapplicationsof Orbinstudyingorbifoldsandinthe classicationof knottedgraphs. Hyperbolic invariants are usedtodistinguish the graphs in Litherlands table of 90 prime -curves and provide accesstonewtopological informationincludingsymmetrygroups. Thenbyprescribingconeanglesalongtheedgesofknottedgraphs, tablesoflowvolumeorbifoldsareproduced.iDeclarationThisistocertifythat(i)the thesis comprises only my original work towards the PhD except whereindicatedinthePreface,(ii)due acknowledgement has been made in the text to all other material used,(iii)thethesisislessthan100,000wordsinlength, exclusiveoftables, maps,bibliographiesandappendices.DamianHeardiiiPrefaceSection1.1and 1.2largelyreviewknowntheory. Section1.3 givesnewresults,buildinguponpreviousworkofThurston([61])andUshijima([66]).Section 2.1 gives basic background on orbifolds. Section 2.2, 2.3 and 2.4 producenew results extending the work of Casson ([12]). In Section 2.5, the work of FrigerioandPetronio([23])istranslatedintothesettingofSection2.2.Section3.1wasinspiredbytheworkofScharlemannandWu([58])andLack-enby([41])whileSection3.2and3.3consistentirelyoforiginalwork.InSection4.1theknottedgraphsin[43] and[47] aredistinguishedusinghy-perbolic invariants. Section4.2produces newtables of lowvolume hyperbolic3-orbifolds.TheAppendixoutlinesanalgorithmfortriangulating3-orbifoldsbasedontheauthorsHonoursproject[30].vAcknowledgementsI would like to expressmy gratitude to my supervisor Craig Hodgson. The lastthreeandahalfyearshavebeenafantasticlearningexperiencemadepossiblebyhispatienceandsupport.I alsothankthemanyother peoplewhohavehelpedmealongthe way: OliverGoodmanforinsightfuldiscussionsoncomputerprogrammingandhyperbolicge-ometry; MartinScharlemannfor informative correspondence onhis paper [58];MorwenThistlethwaitewhowasonlytoowillingtogiveadviceandassistancere-gardingthedevelopment of Orb; andJeWeeks fortheopportunitytousehisground-breakingcomputerprogramSnapPea.Finally, I thankmygirlfriendAlice. I havereliedheavilyonher continuousencouragementandsupportduringthisintenseperiod.viiContentsAbstract iDeclaration iiiPreface vAcknowledgements viiListofFigures xiListofTables xvNotation xviiIntroduction 1Chapter1. GeneralizedtetrahedraandtheirGrammatrices 51.1. Hyperbolicspace 51.2. Generalizedtetrahedra 91.3. Grammatrices 11Chapter2. Findinghyperbolicstructureson3-orbifolds 172.1. Orbifolds 172.2. Theparametersandequations 212.3. Flatandnegativelyorientedtetrahedra 292.4. Paredmanifolds 352.5. Canonicalcelldecompositions 372.6. Furtherextensions 45Chapter3. Attaching2-handles 473.1. Boundsonexceptionalcurves 483.2. Thealgorithm 553.3. Twosimpleexamples 583.4. Remarksonimplementation 66Chapter4. Applications 694.1. Knotted-curves 694.2. Lowvolumehyperbolic3-orbifolds 79ixx CONTENTS4.3. Futureapplications 81Bibliography 87AppendixA. TriangulatingorbifoldsoftypeQ = (S3, ) A.91ListofFigures1.1 LorentzianspaceE1,n. 61.2 Thesigneddistancefromahyperplanetoahorosphere. 71.3 ThepictureinP21. 91.4 Vertextruncation. 101.5 Alength-0edge. 111.6 Atetrahedroninscribedinarectangularbox. 152.1 Amodelfromrotationalsymmetry. 182.2 Amodelfromreections. 182.3 Theopensets |coverXQ. 192.4 The2-orbifoldsT(3)andS2(2, 3, 4). 192.5 The2-orbifoldS2(2, 2, 2, 2) isEuclidean. 202.6 Graphsgiveorbifolds. 212.7 Inthelinkofthecuspthepreferredhorospherical trianglesmatch. 232.8 Anexampleorbifold. 242.9 Anexampletriangulation. 242.10 Non-uniquenessofsolutions. 272.11 Someattetrahedra. 302.12 Moreattetrahedra. 302.13 Detectingnegativelyorientedtetrahedra. 322.14 Therighthandrule. 332.15 Theboundaryofaparedmanifold. 352.16 Labellingatrivalentgraph producesaparedmanifold. 362.17 Slicingoanedge. 372.18 Thethreetotwomove. 412.19 Thepositioningof. 433.1 A2-handleaddition. 47xixii LISTOFFIGURES3.2 Coplanarcurves. 483.3 Asg ,c(Sg) = O(log(g)). 493.4 PullingPacrossthediscD. 513.5 Boundarycompressiongivesanewsurfaceoflowercomplexity. 513.6 TwopossiblepictureswhenPisatorus. 523.7 SpinningtheedgesofTaroundP. 533.8 ThepictureintheuniversalcoverofM. 533.9 Anormalcurve. 553.10 Thesubdivisionofneighbouringtetrahedra. 563.11 Aftersubdivision. 563.12 Ungluingfaces. 573.13 Adierentviewofthechasm. 573.14 Slidingtetrahedradownthechasm. 583.15 Theknotted-Y (1. 583.16 Thegluingpatternfor /1. 603.17 Ashortmeridian. 603.18 /2isthecomplementof (2inS3. 613.19 Thegluingpatternfor /2. 623.20 Ashortseparatingcurve. 623.21 Estimatingthedistancebetweenbasepoints. 633.22 Calculatingm. 643.23 Apiecewisegeodesicpath. 643.24 Expandingballsaroundtheneighboursofx. 654.1 Twocompositegraphs. 694.2 Thesmallestcuspedorientablehyperbolic3-orbifoldandthesmallestknownorientablehyperbolic3-orbifold. 794.3 Thethreesmallest orientablehyperbolic3-orbifoldswithnonrigidcusps. 794.4 Enumeratingknottedgraphs 80A.1 Truncatingtheverticesandthenshrinkingedgesof(Q). A.91A.2 isinS3. A.92A.3 CuttingupS2I. A.92LISTOFFIGURES xiiiA.4 Oneofthesecondtypesofpieces. A.93A.5 CuttingupS2I. A.94ListofTables3.1 Thelistofexceptional curveson/1(uptosymmetry)oflength c(S2). 593.2 Thelistofexceptional curveson/2(uptosymmetry)oflength c(S2)andwhoseintersectionwithisessential. 614.1 TheorbifoldsofthetypeQ = (S3, )foundwithvol(Q) &lt; 0.5,whereisaconnected,prime,trivalenttwovertexgraphwithatmost7crossings. 824.2 Somesimpleprime,trivalenttwovertexgraphs. 834.3 TheorbifoldsofthetypeQ = (S3, )foundwithvol(Q) &lt; 0.2,whereisaconnected,prime,trivalentfourvertexgraphwithatmost7crossings. 844.4 Somesimple,prime,trivalentfourvertexgraphs. 85xvNotationSymbol MeaningEnEuclideann-spaceHnHyperbolicn-spaceSnSphericaln-spaceE1,nLorentzian(n + 1)-space, ) TheLorentzianinnerproductHThehyperboloidmodelof HnPn1Theplanex0= 1inE1,nT(x) Theradialprojectionofx E1,ntoPn1BnTheopenunitballinPn1 AtetrahedronAgeneralizedtetrahedroninE1,3 AgeneralizedtetrahedroninH3viThei-thvertexofinE1,3vijvi, vj)V ThematrixwiththeverticesviofascolumnsJ Thediagonalmatrixdiagonal(1, 1, 1, 1)G ThevertexGrammatrixof= VtJV= (vij)wiAnormaltothei-thfaceofinE1,3wijwi, wj)W ThematrixwiththenormalswiofascolumnsGThenormalGrammatrixof= WtJW= (wij)GijThematrixobtainedbydeletingthei-throwandj-columnfromGcijThe(i, j)-thcofactorofG = (1)i+jdet(Gij)xviiIntroductionTheclassicationof2-manifoldsissomethingwell understood. Theclassica-tion of 3-manifolds is a much harder problem. We do not even have conjectural listofall3-manifolds.IfThurstonsGeometrizationConjectureisconrmed, whichseemsmoreandmorelikelyduetotheworkof Perelman, thenwewouldhaveacompletesetoftopological invariants. Inparticular, for irreducible atoroidal 3-manifolds, withthe exception of lens spaces, the fundamental group would be a complete invariant.Unfortunately the fundamental group alone does not provide us a practical methodofdistinguishing3-manifolds.Tothisend, topologistshavebeenrelyingheavilyongeometrytodistinguishbetween3-manifolds. Ageometricstructureonamanifoldisacomplete, locallyhomogeneousRiemannianmetric. Inparticular, ahyperbolicmanifoldisaRie-mannianmanifoldwithconstantsectional curvature 1. Hyperbolic3-manifoldsare the most interesting, and most abundant, while non-hyperbolic 3-manifolds arelargelyunderstood.In[61], ThurstonintroducedhyperbolicDehnsurgery, amethodforcontinu-ously deforming the topology and geometry of a hyperbolic 3-manifold to a dierent3-manifold. ThecomputerprogramSnapPea([69]), developedbyWeeks, allowstheusertoexplorethisprocess. Manifoldsarespecial examplesof spacescalledorbifolds, whicharemodelledlocallyonRnmodulonitegroupsof symmetries.Onegoal of thisthesisistoextendtheideasusedinSnapPeatotheclassof 3-orbifolds. TheseconceptsareimplementedinacomputerprogramOrb. AswithSnapPea, Orbshouldprovideinvaluableinformationonhyperbolic3-orbifoldsandaidfuturetheoreticalwork.Therstchapterisareviewofsomehyperbolicgeometryandadiscussionofgeneralizedtetrahedra. Generalizedtetrahedraarisewhenweallowtetrahedrathat have vertices at and beyond the boundary of 3-dimensional hyperbolic spaceH3. Combinatorially,ageneralizedtetrahedronisjustatetrahedronwithsomeofitsverticesslicedo. SuchatetrahedroncanberealizedgeometricallyinH3byslicinganyhyperinnitevertices oalongtheircorrespondingdual hyperplanes.See Section1.2for moredetails. We canuse thehyperboloidmodel of hyper-bolicspacetopositionanygeneralizedtetrahedroninLorentzianspaceE1,3. If12 INTRODUCTIONv1, v2, v3, v4 E1,3are the vertices of , then the vertex Gram matrix of is thesymmetric44matrixofLorentzianinnerproductsG=(vi, vj)). ThematrixGcompletelydeterminesuptoisometryandsoitcanbeusedtorecoveritsdihedralanglesandedgelengths.In [61], Thurston devised a way of subdividing the gure-eight knot complementinto two regular ideal hyperbolic tetrahedra. Weeks has drawn upon this approachtodevelopthecomputerprogramSnapPeawhichcansubdividethecomplementof alinkinS3intoideal tetrahedra. It canthensearchfor tetrahedrasothatthesumofthedihedral anglesaroundeachedgeinthetriangulationis2. Thisdeterminesahyperbolicstructureonthemanifold, givingaccesstoavastarrayof geometric invariants. Cassonhas alsodevelopedaprogramGeo([12]) thatcomputes geometric structures onclosed3-manifolds bysubdividing themintonite tetrahedra. Although both these programs have proven invaluable in studying3-manifolds,theyarelimitedbythekindoftetrahedratheyuse.Thurstonalsosuggestedthat thismethodcouldbe extendedtoworkon graphcomplements. Heshowedin[62] thatthecomplementoftheknottedYcouldbesubdivided into two regular generalized tetrahedra. Frigerio and Petronio proposedonewayof implementingthisapproachin[23] usingthedihedral anglesof thegeneralized tetrahedraas parameters. This has been implementedwith Martelli inthecomputerprogramographs([21]).Thesecondchapterdevelopsanalternativemethodforparametrizinggeneral-izedtriangulations, usingvertexGrammatricesof thegeneralizedtetrahedraasparameters in an approach similar to that of Casson in Geo. The shapes of the gen-eralized tetrahedra in a triangulation Tcan be completely determined by [T0[+[T1[parameters,where [Ti[ is the numberof i-cells in T, signicantlyfewer parametersthanrequiredbytheapproachin[23].This techniquefor ndinghyperbolic structures can also be used on closed andcusped3-manifoldsandon3-manifolds withgeodesicboundary. It canalsobeusedtondstructures onaverylargeclass of 3-orbifolds. Wecandothis byrelaxingtheedgeconditionbyallowingtheconeanglearoundeachedgetobe2n ,forsomen 1. Sinceorientable3-orbifolds looklikeorientable3-manifoldswithembeddedtrivalentgraphsassingularloci, averylargeclassof orbifoldscanbedealtwithinthisway.Orbisacomputerprogramwhichimplementsthismethodforparametrizingtriangulations. It canstart withaprojectionof agraphembeddedinS3, andproduceandsimplifyatriangulationwithsome prescribedsubgraphas part ofthe 1-skeletonandthe remainder of the graphdrilledout. (This is describedINTRODUCTION 3intheAppendix.) Orbthenuses thevertexGrammatrices toparametrizethetriangulationandsolveforahyperbolicstructureusingNewtonsmethod.Givenacompact3-manifold Mandan essentialsimpleclosedcurveonM,wedeneM[] tobethemanifoldobtainedbygluinga2-handletoMalong. If liesonatorusboundarycomponent, wecapothespherical boundarycomponentcreatedandtheresultisjustDehnlling.SupposeT is atorus boundarycomponent of M, T, andsupposeMishyperbolic. ByThurstonsHyperbolicDehnSurgeryTheorem([62]), thereareonly a nite number of slopes with non-hyperbolic M[]. Thurston and Gromov([28],[6])alsoshowedthatifthelengthof,asmeasuredintheEuclideanmetriconthe boundaryof ahoroball neighbourhoodof the cusp, is at least 2thenM[] is negativelycurved. Agol ([3]) andLackenby([42]) have independentlyshownthat if the lengthof (measuredas above) is at least 6thenM[] isirreducible, atoroidalandnotSeifertbered,andhasaninnite, wordhyperbolicfundamentalgroup. Hodgson and Kerckho ([32]) have shownthat the numberofnon-hyperbolicllingsisboundedbyanumberindependentofM.The third chapter examines the case when lies on a boundary surface of genus 2. Using a lemma of Scharlemann and Wu ([58]), an argument of Lackenby ([41])and a theorem on the density of strip packings, due to Marshall and Martin ([44]),thefollowingresultisproven.Theorem3.1 Let Mbeanorientablecompact nitevolumehyperbolic3-manifoldwithnon-emptygeodesicboundary. SupposeisasimpleclosedgeodesiconaboundarycomponentS,withgenusgreaterthanone. Letc(S) = 6ArcCosh___1 +2_1 4/(S)__1 4/(S) 1_2___.ThenM[] ishyperbolicprovidedthat,ifisseparatingthenLength() &gt; c(S),andifisnon-separating,thenall curvescoplanartohaveLength() &gt; c(S).Two curves and on surface Sare coplanarif some component of S()isanannulusora3-puncturedsphere. IfMishyperbolicandM[]isnottheniscalledanexceptional curve.Thestudyof2-handleadditionsconcludesbyenumeratingexceptional curvesontheboundaryof twoof theeight lowest volumehyperbolic3-manifoldswithgeodesic boundarydeterminedbyFujii in[24]. This is done byproducinganalgorithmwhichstartswithatriangulated3-manifoldMwithacurve M4 INTRODUCTIONand creates a triangulation for M[]. These examples turn out to be very dierent,onehavinganitelistandtheotheraninnitelistofexceptionalcurves.The enumerationandclassicationof knots andlinks has benetedgreatlyfromtheinformationthathyperbolicstructuresprovide. Mostow-Prasadrigidityimpliesthat a completehyperbolic structure is a completeinvariant of a nite vol-ume hyperbolic 3-manifold ([48],[52]). This result means that knots and links withhyperboliccomplementscanbedistinguishedbytheirgeometricstructures. Hy-perbolic structurescan be used in a similar fashion to distinguish between knottedgraphsinS3.A-curveisaspatial graphinS3consistingof twoverticesandthreeedges,whereeach edge joins the twovertices. Litherland ([43]) and later Moriuchi ([47])enumeratedall prim...</p>