On Floer homology and the Berge conjecture on knots admitting lens space surgeries

Abstract

Abstract. We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge's construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a threesphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.

Dehn surgery, rational open books and knot Floer homology

“…This is a rational analogue of the Heegaard diagram for fibered knots constructed in [OS6], and may be useful for understanding the interaction between properties of the monodromy of a rational open book and those of the contact invariant. By combining the theorem with results of [Ni2] and [He4] (see also [Ra]) we arrive at the following corollary.…”

Surgery obstructions from Khovanov homology

“…16 For generic associated quotient tangles, w min and w max provide lower and upper bounds respectively for w(τ ( p q )) for any reduced p q ∈ Q. Proof The upper bound as claimed is established in Sect.…”

Knots which admit a surgery with simple knot Floer homology groups

“…We may now prove the following theorem, which should be compared with Theorem 1.2 of [12] and Theorem 2.4 from [7]. Theorem 4.1 Let K X be a knot in a homology sphere L-space X of Seifert genus g.K/, and fix the above notation.…”

“…The importance of studying knots inside rational homology spheres which have simple knot Floer homology came up in the study of the Berge conjecture using techniques from Heegaard Floer homology by Hedden [7], Rasmussen [14] and Baker, Grigsby and Hedden [1]. By definition, a knot K inside a rational homology sphere X has simple knot Floer homology if the rank of b HFK.X; K/ is equal to the rank of b HF.X / (see Oszváth and Szabó [11; 10] and Rasmussen [15] for the background on Heegaard Floer homology and knot Floer homology).…”