Combinatorial Description of Knot Floer Homology of Cyclic B(4)

branchingaroundthe2npoints.

Forthem-foldcycliccoverthisyieldsa2n-pointedgenus(m 1)n+1Heegaarddiagram Σm(K).withmnαcurvesandmnβcurves,yieldingmn!generatorsfortheforK

HeegaardFloerchaincomplex.ThesymmetriesinthischaincomplexmaybeexploitabletomakeitshomologynearlyasfasttocomputeastheoriginalS3knotFloerhomology. (S,(α,Jα),(β,Jβ),w,z;s1).Figure14.Generatorsanddi erentialsinCF

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

20J.ELISENDAGRIGSBY

Weshouldremarkthatinthiscasethefundamentaldomainscontainingthebasepointswillallbe2mn-gons,sowewillonlyhavecombinatorialaccesstoinformationaboutthe

associatedgradedcomplex,HFK.

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E-mailaddress:egrigsby@math.columbia.edu