Grid diagrams and Legendrian lens space links

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 , and they have a straightforward connection to Legendrian representatives of K (S 3 , st ), where st is the standard, tight contact structure. The definition of a grid diagram was extended, by Hedden and the authors, 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.