In this paper, we introduce a sequence of invariants of a knot K in S 3 : the knot Floer homology groups HFK(Σ m (K); K, i) of the preimage of K in the m–fold cyclic branched cover over K. We exhibit HFK(Σ m (K); K, i) as the categorification of a well-defined multiple of the Turaev torsion of Σ m (K) − K in the case where Σ m (K) is a rational homology… (More)
In , Lawrence Roberts, extending the work of Ozsváth and Szabó in , showed how to associate to a link, L, in the complement of a fixed unknot, B ⊂ S 3 , a spectral sequence whose E 2 term is the Khovanov homology of a link in a thickened annulus defined in , and whose E ∞ term is the knot Floer homology of the preimage of B inside the… (More)
Until recently, individuals with premutation alleles (55-200 CGG repeats) of the fragile X mental retardation 1 (FMR1) gene were believed to be psychologically unaffected. However, the recent documentation of abnormal elevation of FMR1 mRNA, discovery of fragile X-associated tremor/ataxia syndrome (FXTAS), and reports of psychiatric disorders in children… (More)
In this paper, we introduce a simple combinatorial method for computing all versions (∧, +, −, ∞) of the knot Floer homology of the preimage of a two-bridge knot Kp,q inside its double-branched cover, −L(p, q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted version of the toroidal grid diagrams recently introduced by Manolescu,… (More)
In , Ozsváth-Szabó established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L ⊂ S 3 and the Heegaard Floer homology of its double-branched cover. This relationship, extended in  and , was recast, in , as a specific instance of a broader connection between… (More)
We show that the sutured Khovanov homology of a balanced tangle in the product sutured manifold D × I has rank 1 if and only if the tangle is isotopic to a braid.