Combinatorial Description of Knot Floer Homology of Cyclic B(2)

Morespeci cally,Section2.3of[OS04b]andSection2.6of[OS04d]),describehowtosplitthechaincomplexCF K(Y;K)associatedtoatraditional2-pointedHeegaarddiagramintosubcomplexesindexedbySpinc(Y0(K)).Furthermore,theyspecifyamap

f:Tα ∩Tc

β →Spin(Y0(K)),

assigninggeneratorsofCF K(Y;K)toSpinc(Y0(K))structures.

Theygoontoconstructasplitting

p1×p2:Spinc(Y0(K))～=

)onacappedo SeifertsurfaceF fortheknot:

p2(s2<c1(s

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

6J.ELISENDAGRIGSBY

IfYisarationalhomologysphere,themapp2isindependentofthechoiceofSeifertsurfacefortheknot.

TheabsoluteAlexandergradingisthenuniquelydeterminedforageneratorxinrelativeSpincstructures

2<c1(s

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

CYCLICBRANCHEDCOVERS7

x

Figure2.A0 1,2 3HeegaarddiagramstabilizationforaknotK, wobtainedbyreplacingasmallneighborhoodofapointpinS α β z

nearzwiththeabovelocalpicture.

pnearz(whereby“nearz”wesimplymeanthatwecanconnectztopbyanarconSwhichdoesnotintersectanyαorβcurves).

(3)NotethatthenewchaincomplexsplitsasthedirectsumCx⊕Cyoftwocopiesofour

oldchaincomplexwhereCxconsistsofthosegeneratorsoftheform( ,x)andCy

consistsofthosegeneratorsoftheform( ,y).Furthermore,twogeneratorsofthis

chaincomplexwhichagree everywhere exceptinthelastcomponentareconnected

byanobviousdiskφwithnzi(φ) nwi(φ)=1andµ(φ)=1.Iteratingthis

processyieldsthedesiredconclusion.

w,z)naturallygivesrisetoa (S,αAsmentionedearlier,theAlexandergradingonCF ,β, (K,m) Z ltration.Inparticular,(seeSection3.1of[MOS])onede nessubcomplexesF w,z)generatedbyelementsUa2···Uanxwith (S,αCF ,β,n2

a2A(U2anx)···Un=A(x) n i=2ai≤m.

De nition2.4.IfYisanL-space,τs(K)isde nedtobetheminimalmforwhichthemapinducedonhomology

w,z) i :H (F ,β,s(K,m))→HF(S,α s(K,m)This ltrationsplitsnaturallyintoSpinc(Y)buckets.Accordingly,onedenotesbyFthesubcomplexgeneratedbyelementsinswithA(x)≤m.

isnon-trivial.

Remark:τs(K)isde nedinfargreatergenerality(seeSection5of[OS03b]),butwerestricttothisspecialcaseforthesakeofexposition,sinceitisallwewillneedatpresent.

p,q)3.NiceHeegaardDiagramsfor(Σ2(Kp,q),K

Inthissection,weexplicitlyconstructthe4-pointedtwistedtoroidalgriddiagramfor p,q Σ2(Kp,q)describedintheintroduction.K

Onceagain,ourconventionistodenotebyKp,qthetwo-bridgelinkwithdouble-branchedcover L(p,q).

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

8J.ELISENDAGRIGSBY1

2

3

...

cn

Figure3.Atwo-bridgelinkwithcrossingnumbers(c1,c2,...,cn).Posi-

tive=RH,andNegative=LH.

Recall(see,e.g.,Chapter12of[BZ03])thatatwo-bridgelinkLhastheformgiveninFigure3.Itisastandardfactthattwo-bridgelinksareclassi edbytheorientedhomeo-morphismclassoftheirdouble-branchedcovers,thelensspacesL(p,q)(whereweassumethatp∈Z+,q∈Z,and(p,q)=1).Moreprecisely:

Theorem3.1.[Sch56,Con70,Rei35,Bro60]Atwo-bridgelinkwithcrossingnumbers

(c1,c2,...,cn)

hasdouble-branchedcovering L(p,q),where

p

c2 1

··· 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

CYCLICBRANCHEDCOVERS9

Side view of K

Top view (after K falls onto S)

Figure4.ObtainingtheSchubertnormalformofK7,3

w

Figure5.4-pointedHeegaarddiagramscompatiblewiththeknotsK3,1

andK5,2,respectively

FromtheSchubertnormalformforKp,qweconstructagenus0,four-pointedHeegaarddiagramforS3compatiblewithKp,q.

InthecasewhereKisthetwo-bridgeknotKp,q,weidentifythemaximaofthetwoupperbridgeswiththetwoindex0criticalpointsoff(labeleda1anda2)andtheminimaofthetwolowerbridgeswiththetwoindex3criticalpointsoff(labeledb1andb2).ThentheαcurveisaregularneighborhoodofeitheroverbridgeandtheβcurveisaregularneighborhoodofeitherunderbridgeintheSchubertnormalformforKp,q.Ifqisodd,thenthepoints(readfromlefttoright)arew1,z1,z2,w2,whileifqiseven,thenthepoints(readfromlefttoright)arew1,z1,w2,z2.SeeFigure5.

Sinceitistraditionalintheliteraturefortheαcurvestobesimpleandtheβcurvescomplicated,wewillstraightenouttheαcurvebyperforminganisotopyoftheαandβcurvesavoidingthewiandzi,whichhasthee ectofre ectingthepictureaboutahorizontalaxisandswappingtherolesofαandβ.Alternatively,wecouldhaveusedthemirror,Kp, q,andswitchedtherolesofαandβbeforelettingtheknotfallontoStogetaHeegaarddiagramcompatiblewithKp,q.SeeFigure6.

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

10J.ELISENDAGRIGSBY

w

Figure6.4-pointedHeegaarddiagramcompatiblewithK3,1after

straighteningouttheαcurve

zLeft: q is oddRight: q is even

Figure7.TheintersectionofaSeifertsurfacewithS(indicatedbythe

dottedline)inthecasewhereqisodd(left)andeven(right)

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