Combinatorial Description of Knot Floer Homology of Cyclic B(3)

12J.ELISENDAGRIGSBY

3.2.Alexandergrading.Notethattheorientedknotitselfcanbeseeninthediagramasapiecewiselinearunionofhorizontalarcsandarcsofslopep

completestheproof.

OneliftstoanabsoluteAlexandergradingbymakingtheuniquechoiceyieldinganappropriatelysymmetricEuler

characteristic2PD(c1(sx) c1(sy))

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

CYCLICBRANCHEDCOVERS13

χ(HF K(Y;K,m)):=rk(HFKd(Y;K,m))

d

where ( 1)d ∈Z2

HF K(Y;K,m):=

{s<c )21(s

isthesubcomplexofHF K(Y;K)consistingofthoseSpincstructures

evaluationonthehomologyclassofacapped-o SeifertsurfaceF withtheappropriate

fortheknot.

Moreprecisely,recall(seeProposition3.10of[OS04b])thatthereisaconjugationsym-metryonHF K(Y;K)forKanullhomologousknotinarationalhomologysphere.Thisimplies:

Corollary3.1.

K(T):=

polynomialin i∈Zχ(HF K(Y;K,m))·Ti

isasymmetricLaurentT.

whichallowsusto xtheabsoluteAlexandergradingbymakingtheuniquechoicewiththeproperty

3.3.Spincgradings.We TA(x)T:= K(T)·(1 T 1)n 1x∈.α ∩Tβ canmakesimilarlocalassignmentstopartitionthegeneratorsintoSpinc( L(p,q))structures.

Moreprecisely,forthetwistedgriddiagramS

→ associatedtoKZp p,q L(p,q)wede neamapS:X

fromintersectionpointstointegersmodp.Beforedescribinghowtodothis,recallouridenti cationinSection1of

(1)S

(2)α withthefundamentaldomain[0,1]×[0,1],andJ(α)withtheimagesinS =R2/Z2ofthelinesy=0(=1)andy=1

qxandy=p

2)respectively.

Nownoticethattheverticesofthetwistedgriddiagramareat(j

2p,1

β

αp,0)2p+j

J(α)2p+j12)p,

p,0)=j∈Zp

(2)S(1p,0)=j∈Zp

(3)S(j

2)=j+q 1

j=j+q+1

2p+2)

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

14J.ELISENDAGRIGSBY

whereasinthecaseq<0wemaketheassignment:

(1)S(j

j

2p+

1p,2∈Zp

1(4)S(1p,2∈Zp

SeeFigure11inSection4fortheexampleofK3,1.

(T)whosecomponentsaretheintersectionProposition3.2.LetxbeageneratorofCF

pointsx1andx2.Thefunction

S(x):=2 i=1S(xi)∈Zp

w,z)intoSpinc(Y)structures,si,indexedbyele- (S,αpartitionsthegeneratorsofCF ,β,

mentsofZp. sendingα J(α)andFurthermore,ifJ(x)istheimageofxundertheinvolutiononS

β J(β)andx∈si,then

J(x)∈s i.

TheJ-invariantgeneratorsarepreciselytheonesins0.

ProofofProposition3.2First,recallthatSpinc( L(p,q))isana nesetfortheactionof

H1( L(p,q))～=Zp.

Furthermore,sincepisodd,thereisauniquespinSpincstructure,anditisnaturaltoidentifythisSpincstructurewiththe0elementofZp.Thischoiceallowsus,oncewehavechosenageneratorγ∈H1( L(p,q)),toidentifytheSpincstructureswithZpbycomparingc1(s)withγ.

Moreprecisely,ifPD(c1(s))=2m·γ,thenweidentifyswithm∈Zp.

(T)inthedesiredfashion,ToseethatthegivenfunctionpartitionsgeneratorsofCF

beginbynoticingthattheJ-invariantgenerators,x=(x,J(x)),arepreciselytheonesforwhich2 S(xi)=0∈Zp,

i=1

sinceJcanbedescribedasthe180degreerotationinR2aboutthecenterofthecellcon-tainingthebasepointwintheupperleft-handcorner2andwehavechosenourassignmentsStobeanti-symmetricwithrespecttothisoperation.I.e.,

S(xi)≡ S(Jxi)modp.

Thisprovesthat

(x1,x2)isJ-invariant=

c2 i=1S(xi)=0.TomeasuretheSpinstructureofanynonJ-invariantgeneratory=(y1,y2), rstchoose

acurveγonT,representingahomologyclassonTwiththeproperty

<γ,β>=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

CYCLICBRANCHEDCOVERS15

tobeourgeneratorofH1( L(p,q)).By< , >,wemeanthestandardintersectionpairingonH1(T).

Thensimplyconnectytoy0:=(y1,Jy1)∈s0bypathsalongtheαandβcurvestoform (y,y0).NoticethatourassignmentSwasmadesothat (y,y0)measurestheintersectionpairingwithβ:

S(y)=< (y,y0),β>,

yieldingourchosenisomorphismofH1( L(p,q))withZp.

3.4.AbsoluteZ2andQhomologicalgradings.WecanspecifytheabsoluteZ2homo-logicalgradingongeneratorsbyperformingasimilarlocalsum.

Speci cally,recallthatweareassumingthat p<q<p,qisodd,andwearespecifyingtheintersectionpointsintermsoftheircoordinatesinthefundamentaldomain[0,1]×[0,1] R2.Thende ne 1M:X→0,

=M(i2)=0

1p,0)=M(

p,

p,0)112

p,(2)M(p,0)1p,12andifq<0,(1)M(i(2)M(1=M(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

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Figure10.Findinghomologicalgrading0generatorsineachSpincstructure

TheMaslovindexzerotrianglesmentionedintheproofaboveconnectthestandardlensspacegenerators(whoseQgradingsarecalculatedin[OS03a])togeneratorsonourHeegaarddiagram,yieldingaQgradingassignmentforonegeneratorineachSpincstructure.

Thenwenote,asin[MOS],thatifx,yaregeneratorsinthesameSpinc(Y)structureandφ∈π2(x,y)then

M(x) M(y)=Px(φ)+Py(φ) 2·W(φ),

where

Px(φ):=2 i=1p(xi)(φ),

pxi(φ)istheaverageofthelocalmultiplicitiesofφinthefourquadrantsaroundxi(Seethede nitionofnp(A)inSection4.2of[Lip]),and

W(φ):=2 i=1nwi(φ).

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