Manifolds with small Heegaard Floer ranks

Abstract

We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink. 57M27; 57M25

Thurston norm and cosmetic surgeries

“…. Moreover, the small triangle forỹ 1 in Γ 2 is just a translation of the small triangle for y 1 in Γ 1 , so they contribute the same n w (ψ) − n z (ψ) term in (2). So (3) follows.…”

Khovanov module and the detection of unlinks

“…Their result had numerous antecedents [1,2,3,10,11,12,13,37], most aimed at generalizing or exploiting a spectral sequence discovered by Ozsváth and Szabó [31] which begins at Khovanov homology and converges to the Heegaard Floer homology of the branched double cover.…”

Dehn surgery and non-separating two-spheres

“…The key idea in the proof of Theorem 1.1 comes from the special property of the twisted Heegaard Floer homology of three-manifolds with non-separating S 2 's. (This has been used in [Ni09] and [Ni13]; see also [HN10], [HN13], and [AL19].) In the next section, we review the mapping cone formula in Heegaard Floer homology, with extra attention to twisted coefficients, and prove Theorem 1.1.…”