Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Abstract

Abstract. Similar to knots in S 3 , any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S 3 and the similarity of the resulting combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S 3 admitting lens space surgeries.

Khovanov homology, sutured Floer homology and annular links

“…The bigrading referenced in the above proposition is the .A; M/ bigrading associated to the null-homologous link z B †.S 3 ; L/, where A is the filtration (Alexander) grading on knot Floer homology defined by Ozsváth and Szabó in [17] and Rasmussen in [27] and elaborated by Ozsváth and Szabó in [24] and Manolescu, Ozsváth and Sarkar in [13] (see also Ni [15] and Baker, Grigsby and Hedden [3]), and M is the homological (Maslov) grading on Heegaard Floer homology defined by Ozsváth and Szabó in [23]. …”

Grid diagrams and Legendrian lens space links

“…Furthermore, the observation that all Reidemeister moves are local, To prove that any two smoothly isotopic links K 1 and K 2 have topologically gridequivalent grid diagrams, we use the argument outlined by Dynnikov in [Dyn06]. First, we represent K 1 and K 2 by (rectilinear) toroidal projections, using Lemma 4.2 of [BGH07]. By the Reidemeister theorem for lens space links, we can move from one toroidal projection to the other by a sequence of smooth isotopies and moves of type I -IV.…”

“…By the Reidemeister theorem for lens space links, we can move from one toroidal projection to the other by a sequence of smooth isotopies and moves of type I -IV. By the same arguments used in the proofs of Lemma 4.2 and Proposition 4.3 of [BGH07], we can approximate each stage of this process using a grid diagram. Provided that each intermediate step is sufficiently simple (subdivide the compact isotopy further if not), it is easy to verify that each step can be accomplished using elementary grid moves.…”

“…Each generator, x, furthermore comes equipped with a combinatorially-defined homological grading, M(x) ∈ Q, as described in Section 2.2.2 of [BGH07]. Now let z − be the generator of CF − (G K ) in the SW (lower left) corner of the X basepoints and z + be the generator of CF − (G K ) in the NE (upper right) corner of the X basepoints.…”

Knots which admit a surgery with simple knot Floer homology groups

“…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).…”