Quantum Relatives of Alexander Polynomial31
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Quantum Relatives of Alexander Polynomial31
th；tha；:v；Xr；aQUANTUMRELATIVESOFALEXA；)(tk,and；theConwayfunction?(L1,..；formulas?k)isarationalfu；tk.Theyarerelatedby(L)(t；k),ifk&1and；?(L)(t)=?L(t2)；1SomeauthorscallittheAle；2?t
2 2 rpA42 ]TG.tham[ 1v 924 2 /tham:viXraQUANTUMRELATIVESOFALEXANDERPOLYNOMIALOLEGVIROUppsalaUniversity,Uppsala,SwedenPOMI,St.Petersburg,RussiaAbstract.ThemultivariableConwayfunctionisgeneralizedtoorientedframedtrivalentgraphsequippedwithadditionalstructure(coloring).Thisisdoneviare?nementsofReshetikhin-Turaevfunctorsbasedonirreduciblerepresentationsofquantizedgl(1|1)andsl(2).ThecorrespondingfacestatesummodelsforthegeneralizedConwayfunctionarepresented.IntroductionAlexanderPolynomialandConwayFunction.TheAlexanderpolynomialisoneofthemostclassicaltopologicalinvariants.Itwasde?ned[1]asearlyasin1928.ArecentpaperbyFintushelandStern[5]hasagaindrawnattentiontotheAlexanderpolynomialbyrelatingittotheSeiberg-Witteninvariantsof4-dimensionalmani-folds.TheAlexanderpolynomialcanbethoughtofinmanydi?erentways.Thereareahomologicalde?nitionviafreeabeliancoveringspaces,ade?nitionviaFox’sfreedi?erentialcalculus,ade?nitionviaReidemeistertorsion,andadiagrammaticalConway’sde?nition.SeeTuraev’ssurvey[18].Asamatteroffact,Conway[3]hasenhancedtheoriginalnotionbyeliminatinganindeterminacy.HeintroducedalsoalinkinvariantwhichencodesthesameinformationastheenhancedAlexanderpolynomial,butismoreconvenientfromseveralpointsofview.Conwaycalleditapotentialfunctionofalink.FollowingTuraev[18],wecallittheConwayfunction.1Thede?nitionsofthe(enhanced)AlexanderpolynomialandConwayfunctionaregivenbelowinSection37.7.Hereletmerecallonlytheirformalappearance.ForanorientedlinkL?SwithconnectedcomponentsL1,...,Lk1theAlexanderpolynomial?L(t1,...,tk)isaLaurentpolynomialinvariablest2)(tk,andtheConwayfunction?(L1,...,tformulas?k)isarationalfunctioninvariablestk.Theyarerelatedby(L)(t1,...,tk)=?L(t21,...,t2t1,...,k),ifk&1and?(L)(t)=?L(t2)1SomeauthorscallittheAlexander-Conwaypolynomial,seee.g.[8],[13]and[14].Howevertherearetworeasonsnottousethisterm.First,thisisnotapolynomialeveninthesenseinwhichtheAlexanderpolynomialis:thisisnotaLaurentpolynomial,butratherarationalfunction.Say,fortheunknotitis12?t?12OLEGVIROAlexanderPolynomialandQuantumTopology.Theinterventionofthequan-tum?eldtheoryintothelow-dimensionaltopology,whichwasinitiatedinthemideightiesbydiscoveryoftheJonespolynomial,addedanewpointofviewontheAlexanderpolynomial.ItwasincludedintoseriesofotherquantumpolynomiallinkinvariantssuchastheJones,HOMFLYandKau?manpolynomials.Invariantsofatopologicalobjectwhicharestudiedbythequantumtopologyarepresentedbyexplicitformulasinspiredbythequantum?eldtheory.Aversatilityoftheseformulasallowsonetogeneralizetheinvariantstowiderclassesofobjectsand?ndcounterpartsoftheinvariantsforcompletelydi?erentobjects.Forexample,theJonespolynomialwas?rstgeneralizedtocoloredlinks,thentotrivalentframedknottedgraphs,thenitscounterparts(Reshetikhin-Turaev-Witteninvariants)ofclosedoriented3-manifoldswerediscovered,andtheywereupgradedtoTQFT’es,i.e.functorsfromcategoriesofclosedsurfacesandtheircobordismswithanappro-priateadditionalstructurestothecategoryof?nite-dimensionalvectorspaces.Theinvariantof3-manifolds(Witten-Reshetikhin-Turaevinvariant)wasde?nedtechni-callyevenasaninvariantof4-manifoldswithboundary,see[20].Althoughitsvaluedependsonlyoftheboundary,theconstructiondealswiththe4-manifold.Tosomeextentthesameschemeworksforanyquantumlinkpolynomial.However,thequantumtheoryoftheAlexanderpolynomialhasnotevolvedtotheextentswhichhavebeenachievedbythetheorybasedontheJonespolynomial.Onlythevery?rststepsofthisprogramhavebeenmade:ithasbeenshownthattheAlexanderpolynomialcanbede?nedviarepresentationsofquantumgroups.ThemultivariableConwayfunctionwasobtainedalongthelinesoftheconstructionoflinkinvariantsrelatedtoaquantumgroupbyJunMurakami[11],[12],RozanskyandSaleur[13]andReshetikhin[15].Rozansky,SaleurandReshetikhinappliedthisconstructiontoquantumsupergroupgl(1|1)(i.e.,aquantumdeformationUqgl(1|1)oftheuniversalenvelopingalgebraofthesuperalgebragl(1|1),Murakamididthiswiththequantumgroupsl(2)atq=√seeAppendixAto[14]).QUANTUMRELATIVESOFALEXANDERPOLYNOMIAL3Inthesl(2)directionthecorrespondingstepsseemtohavebeendonein[4],but,forthesakesofasubsequentdevelopment,wedothemalloveragainfromscratch,ine?orttogetsimplerformulasandgeometricpresentations.Inparticular,thisallowsustocomparetheresultsingl(1|1)andsl(2)directions.Noneofthemcouldbereducedtoanotherone.Coloringsusedingl(1|1)approacharebasedonalargerpalette.Coloringsbasedonapartofgl(1|1)-colorscanbeturnedintosl(2)-colorings,butnotallthesl(2)-coloringscanbeobtainedinthisway.Largerpalettemakesgl(1|1)approachsomehowmore?exible,butleadstomorecomplicatedfacestatesummodels.Bothdirectionsseemtodeserveinvestigation,sincetheymaybeusefulfordi?erentpurposes.TherelativesoftheAlexanderpolynomialstudiedinthispaperare:?theReshetikhin-TuraevquantumfunctorsRT1andRT2basedontheirreduciblerep-resentationsofcgl(1|1)andsl(2),respectively,?amodi?cationAofRTcforc=1,2,?invariants?4OLEGVIROalgebracausesunusualtopologicalfeatures.Forexample,coloringsofgraphsinvolveorientationsof1-stratainmoresubtleway,andatsomeverticesacyclicorderoftheadjacent1-stratamustalsobeincludedintoacoloring.Modi?cationsofReshetikhin-TuraevFunctor.Theobviouspolynomialna-tureoftheBoltzmannweightsinTables1and2andthepolynomialnatureoftheAlexanderpolynomialsuggestamodi?cationofRT1whichgivesrisetoafunctorA1whichactsfromalmostthesamecategoryofcoloredframedgenericgraphstothecategoryof?nite-dimensionalfreemodulesoversomecommutativeringB.ForexampleBmaybeZ[M]whereMisafreeabelianmultiplicativegroup.InthiscaseBistheringofLaurentpolynomials.ThusA1isclosertotheAlexanderpolynomial.Roughlyspeaking,A1isobtainedfromRT1byeliminatingthePlanckconstantq(parameterofthequantumdeformationinthequantizationUqgl(1|1))byreplac-ingthepowersofqwithindependentvariablesintheBoltzmannweights.TheBoltzmannweightsforA1arepresentedinTables3and4.ThetransitionfromRT1toA1doesnoteliminatethequantumnaturetogetherwithq.AlthoughUqgl(1|1)doesnotactintheB-moduleswhicharethevaluesofAonobjects,thereisaHopfsubalgebraofUqgl(1|1)suchthatA1canbeupgradedtofunctorsfromthesamecategorytothecategoriesofmodulesoverthissubalgebra.AlexanderInvariant.TheschemewhichwasusedbyRozanskyandSaleur[13]and[14]forrelatingtheConwayfunctionwiththequantumgl(1|1)versionoftheReshetikhin-Turaevfunctorfortangles,transformsC1totheAlexanderinvariant?1assignstoaclosedcoloredframedgenericgraphΓinR3anelement?1(t21,...,t2k).Thus?QUANTUMRELATIVESOFALEXANDERPOLYNOMIAL5<parewiththerelationbetweentheKau?manbracketsandJonespolynomial.FaceModels.Allformulasofquantumtopologycanbedividedroughlyintotwoclasses:vertexandfacestatesums.Facestatesumsseemtobemoreversatile.Atleast,theyaremoreuniformfortopologicalobjectsofdi?erentnature.Theformulasinthede?nitionoftheAlexanderinvariantareofthevertextype.InthispaperfacestatesumsrepresentingtheAlexanderinvariantarealsoob-tained.Thisisdoneviaaversionof“transitiontotheshadowworld”inventedbyKirillovandReshetikhin[7]toobtainafacestatesummodelforthequantumsl(2)invariantofcoloredframedgraphs,generalizingtheJonespolynomial.Theconstructionusedhereisanotherspecialversionofamoregeneralconstruction,whichIfoundanalyzingKau?man’s“quantumspin-network”construction[6]ofTuraev-Viroinvariants[21],whilepresentingthisinmyUCSanDiegolecturesintheSpringquarterof1991.Ipresentedthegeneralconstructioninseveraltalks,butneverpublished,sinceinthefullgeneralityitlookstoocumbersome,whileinthespecialcases,whichIknewbeforethisworkandinwhichitlooksnice,theresulthadbeenalreadyknown.Thanks.IamgratefultoLevRozanskyforhisvaluableconsultationsconcerningsuper-mathematics,gl(1|1)andpapers[13]and[14]andAlexanderShumakovitchforpointingoutamistakeinapreliminaryversionofthispaper.ContentsIntroduction11.GeometricPreliminariesonKnottedGraphs52.Reshetikhin-TuraevFunctorBasedongl(1|1)93.Reshetikhin-TuraevFunctorBasedonsl(2)214.RelationsBetweentheReshetikhin-TuraevFunctors275.SkeinPrinciplesandRelations326.AlexanderInvariantofClosedColoredFramedGraphs367.SpecialPropertiesofgl(1|1)-AlexanderInvariant398.SpecialPropertiesofsl(2)-AlexanderInvariant469.GraphicalSkeinRelations4910.FaceStateSums5111.Appendix1.Quantumgl(1|1)andItsIrreducibleRepresentations5812.Appendix2.RepresentationsofQuantumsl(2)at√包含各类专业文献、幼儿教育、小学教育、文学作品欣赏、各类资格考试、应用写作文书、高等教育、外语学习资料、中学教育、Quantum 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21 [Honeywell-Tridium 培训教材] Sedona 平台介绍
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相关词典网站：Sedona协议_百度百科
Sedona协议
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对于设备制造商来说：Sedona可以让智能设备的开发更便捷，大大缩短智能设备进入市场的时间（功能强大的编程工具已经具备，大量的潜在客户无需太多的培训和支持）。对于：可以自己为硬件设备开发功能，适应不同用户的要求，并且这个功能还可以移植到新的硬件中。这一切像极了我们今天所使用的智能手机，所有的功能是看第三方的的。
Sedona本质也是一个虚拟机，开源的不仅仅是虚拟机的虚拟引擎，而且还有。对比sedona的虚拟机和Java的虚拟机，发现sedona从Java虚拟机借用的思想还不少，但同时为了能在上很好的运行，也优化了Java虚拟机里面的一些内容，比如：将去掉了，换成了符合图形化开发的架构，这样就非常适合用类似Niagara那样的组件化、图形化的方式来进行业务逻辑的开发。
虚拟机的编写本来就有一定的难度，再重新定义一个sedona语言并实现这个语言的，这就更难了。另外，现在在sedona上面又架构了一个的框架，并实现了sox，以此来完成的图形化开发的工作。即Sedona协议是一套与硬件平台和操作系统无关的，用于构建以网络为中心的智能设备的软件平台架构技术。sedona palette_百度作业帮
sedona palette
sedona palette
第一个单词是一个地名,因此这里音译即可,叫做塞多纳,第二个单词是颜料的意思.
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