Grid diagrams and Legendrian lens space links

Abstract

Abstract. Grid diagrams encode useful geometric information about knots in S 3 . In particular, they can be used to combinatorially define the knot Floer homology of a knot K ⊂ S 3 [MOS06, MOST06], and they have a straightforward connection to Legendrian representatives of K ⊂ (S 3 , ξst), where ξst is the standard, tight contact structure [Mat06,OST06]. The definition of a grid diagram was extended, in [BGH07], to include a description for links in all lens spaces, resulting in a combinatorial description of the knot Floer homology of a knot K ⊂ L(p, q) for all p = 0. In the present article, we explore the connection between lens space grid diagrams and the contact topology of a lens space. Our hope is that an understanding of grid diagrams from this point of view will lead to new approaches to the Berge conjecture, which claims to classify all knots in S 3 upon which surgery yields a lens space.

Morse structures on open books

“…Glue each 1]; the co-core of the handle is the arc {0} × [−1, 1], and we note that the co-core intersects the boundary of the new surface in a pair of points. Similarly, concatenating the core curves [−1, 1] × {0} with rays to the center of the disc produces a wedge of circles onto which the new surface deformation retracts, and we refer to each such loop as a core.…”

“…To use the lemma presented below in the proof above, one needs to choose appropriate minor reparameterizations in the collar neighborhood direction, which is mapped via the Morse function to [0,1] here, but may need to be mapped to some [−a, 0], for example. Also, here the Liouville vector field is presented in the form ∂ ζ instead of, for example, (1 + r)∂ r .…”

“…The fact that any abstract front is the front projection of Legendrian knot follows from the fact that one may directly construct such a Legendrian by flowing via ∓V back into the manifold for a time interval given by the slope on the front. We note that this allows us to easily construct examples, as any curves satisfying the conditions of Definition 1.3 are in fact front projections of Legendrian knots; this is a useful property shared by other instances of front projection (e.g., (R 3 , ξ std ) and universally tight lens spaces) [1].…”

“…Every component of the binding has a standard contact neighborhood, and in fact, this move has already appeared in the literature in the case of front projections of Legendrian knot in universally tight lens spaces. See, for example [1].…”

Computing the Thurston–Bennequin invariant in open books

“…In this section we study rationally nullhomologous Legendrian knots as proposed in Baker-Grigsby [2], Baker-Etnyre [1] and Geiges-Onaran [8]. In particular, we generalise Theorems 2.1 and 3.1 to rationally nullhomologous Legendrian knots.…”