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, Ozsváth, and Sarkar. We conclude with a discuss...

Let f : M → N be a homotopy equivalence between closed negatively curved manifolds. The fundamental existence results of Eells and Sampson [5] and uniqueness of Hartmann [15] and Al’ber [1] grant the existence of a unique harmonic map h homotopic to f . Based on the enormous success of the harmonic map technique Lawson and Yau conjectured that the harmonic map h should be a diffeomorphism. This...

Let L be a link in A×I where A is an annulus. We consider A×I to be embedded in R2×R respecting the obvious fibration and embedding A into a round annulus in R2. We always project L into R2 (or A) along the R-fibration. The complement of L in A × I is thereby identified with the complement of B∪L in S3 where B an unknot as depicted below, called the axis of L. We assume throughout that L inters...

We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers $${{\mathbb {P}}}^3$$ branched along six stable hyperplanes.

Let L ⊂ S be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S, branched along L. When L is an alternating link, ĤF of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E term is a suitable variant of Khovanov’s homology for the link L, convergi...