Combinatorial Description of Knot Floer Homology of Cyclic B

We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted ve

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aCOMBINATORIALDESCRIPTIONOFKNOTFLOERHOMOLOGYOFCYCLICBRANCHEDCOVERSJ.ELISENDAGRIGSBYAbstract.Inthispaper,weintroduceasimplecombinatorialmethodforcomputingallversions(∧,+, ,∞)oftheknotFloerhomologyofthepreimageofatwo-bridgeknotKp,qinsideitsdouble-branchedcover, L(p,q).The4-pointedgenus1HeegaarddiagramweobtainlookslikeatwistedversionofthetoroidalgriddiagramsrecentlyintroducedbyManolescu,Ozsv´ath,andSarkar.WeconcludewithadiscussionofhowonemightobtainniceHeegaarddiagramsforcyclicbranchedcoversofmoregeneralknots.1.IntroductionHeegaardFloerhomology,introducedbyOzsv´athandSzab´oin[OS04d],associatestoaclosed,oriented,connectedthree-manifoldYandaSpincstructures∈Spinc(Y)acollection,HFofgradedabeliangroups, (Y;s),HF∞(Y;s),HF (Y;s),HF+(Y;s),mostnaturallythoughtofastheplexeswithcoe cientsinZ,Z[U,U 1],Z[U],andZ[U,U 1homologygroupsofchaincom-]

TheauthorwaspartiallysupportedbyanNSFPostdoctoralFellowship.

1

We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted ve

2J.ELISENDAGRIGSBY

αβ

J

αα

Figure1.Thetwistedtoroidalgriddiagramwhichisa4-pointedgenus1 7,3 L(7,3)(identifytop-bottom,left-right,inHeegaarddiagramforK

thestandardway).

p,q Inbrief,weobtaina4-pointedgenus1HeegaarddiagramcompatiblewithK

Σ2(Kp,q)whichisatwistedtoroidalgriddiagramconsistingoftwoparallelcurvesofslope0andtwoofslopep

and

thetwocurvesofslopeqxandy=2).Wenowidentify

the

toroidalgriddiagramwiththefundamentaldomain[0,1]×[0,1] R2,1andpositionourfourbasepointsat

11( ,1 ),( ),( )22

where0< <min(1q).SeeFigure1fortheexampleofK7,3.

Following[MOS]weconstructachaincomplex

whosegeneratorsareindexedbybijectionsbetweenthesetofslope0curvesandthesetofslopeppp2

1Notethatby“theimageinT2”wemeantheimageoftheselinesinthequotientR2/Z2andnottheintersectionoftheselineswiththechosenfundamentaldomain.

We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted ve

CYCLICBRANCHEDCOVERS3

InSection3,wedescribetheconstructionofatwistedtoroidalgriddiagramforthedouble-branchedcoverofatwo-bridgeknot.Wealsoshowhowtocombinatoriallyreado Alexander,Spinc,andQMaslovgradings.

InSection4,weperformasamplecomputationofthe lteredchaincomplexinallSpinc

3,1 L(3,1).structuresforK

InSection5,webrie ydiscussgeneralizationsofourmethods.

Acknowledgements:ThispaperwasmotivatedbyjointworkinprogresswithDannyRubermanandSaˇsoStrlewhoseaimistodevelope cientcalculationaltechniqueswithaneyetowardsobtainingnewconcordanceinvariantsforknots.

IamindebtedtoMattHedden,RobertLipshitz,PeterOzsv´ath,DannyRuberman,SaˇsoStrleandJiajunWangformanyinterestingconversationsduringthecourseofthisproject.IamespeciallygratefultoPeterOzsv´athforpatientlyansweringquestionsabout[MOS]andtoDannyRubermanformanyhelpfulcommentsonapreviousdraft.

IwouldalsoliketoremarkthatitwasobservedindependentlybyMattHeddenthatgeneraltwistedtoroidalgriddiagrams(withnparallelcurvesofslope0,nparallelcurvesofslopep

We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted ve

4J.ELISENDAGRIGSBY

w,z)overZ2[U2,...,Un] (S,α[OS,MOS]

associate

tothisdataachaincomplexCF ,β,

freelygeneratedbytheintersectionpointsbetweenthesubvarieties

Tα =α1×...×αg+n 1

Tβ =β1×...×βg+n 1

inSymg+n 1(S)withdi erentialgivenby

(x)=y∈Tα∩Tβ w =(w1,...,wn)and z=(z1,...,zn)aretwon-tuplesofpoints(alldistinct)on ,wherewispeci esaunique owlineγifrombitoai(theonethatintersectsS α βSatwi)andzispeci esaunique owlineηifrombσ(i)toai(σsomepermutationof{1,...,n}).ThenKisuniquelydeterminedbythisdataastheisotopyclassof n γi∪ηii=1and{φ∈π2(x,y)|µ(φ)=1,nw1(φ)=0}# M(φ)

R y.

w,z)thehomologyofthischaincomplex. WedenotebyHFK(S,α ,β,

Remark:Whenn=1,ourK-compatible2-pointedHeegaarddiagramforYyields (Y;K),the lteredchaincomplexarisingintheoriginalformulation[OS04b,Ras03]ofCF

knotFloerhomology.Thehomologyoftheassociatedgradedcomplexofthis lteredchain complex,denotedHFK(Y;K),iscalledtheknotFloerhomologyofKinY,andtheE∞

(Y)obtainedtermofthespectralsequencearisingfromthe ltrationisjusttheordinaryHF

byforgettingthedataoftheknot. w,z)areassignedabigrading(n1,n2)∈Z×Qwhose rst (S,αGeneratorsinCF ,β,

componentiscalledtheAlexander( ltration)gradingandsecondcomponentiscalledtheMaslov(homological)grading.

2.1.Maslovgradings.TherelativeMaslov(homological)gradingbetweentwogeneratorsx,ywithnon-emptyπ2(x,y)isgivenby 2nwi(φ)M(x) M(y)=µ(φ)

i

whereφ∈π2(x,y),

nwi=#(φ∩Vwi)

isthealgebraicintersectionnumberwiththesubvarietyVwi:={wi}×Symg+n 2(S),andµ(φ)istheMaslovindex.< …… 此处隐藏：5337字，全部文档内容请下载后查看。喜欢就下载吧 ……