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.

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

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.

Abstract. In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the Bennequin bound is sharp for fibered knot types. Finally we study rational unknots and show they are weakly Legendrian and transversely simple.In this note we extend the self-linking number of transverse knots and the Thurston-Bennequin invariant and rotation number of Legendrian knots to the case of rationally null-homologous knots. This allows us to generalize many of the classical theorems concerning Legendrian and transverse knots (such as the Bennequin inequality) as well as put other theorems in a more natural context (such as the result in [10] concerning exactness in the Bennequin bound). Moreover due to recent work on the Berge conjecture [3] and surgery problems in general, it has become clear that one should consider rationally null-homologous knots even when studying classical questions about Dehn surgery on knots in S 3 . Indeed, the Thurston-Bennequin number of Legendrian rationally null-homolgous knots in lens spaces has been examined in [2]. There is also a version of the rational ThurstonBennequin invariants for links in rational homology spheres that was perviously defined and studied in [13].We note that there has been work on relative versions of the self-linking number (and other classical invariants) to the case of general (even non null-homologus) knots, cf [4]. While these relative invariants are interesting and useful, many of the results considered here do not have analogous statements. So rationally nullhomologous knots seems to be one of the largest classes of knots to which one can generalize classical results in a straightforward manner.There is a well-known way to generalize the linking number between two nullhomologous knots to rationally null-homologous knots, see for example [11]. We recall this definition of a rational linking number in Section 1 and then proceed to define the rational self-liking number sl Q (K) of a transverse knot K and the rational Thurston-Bennequin invariant tb Q (L) and rational rotation number rot Q (L) of a Legendrian knot L in a rationally null-homologous knot type. We also show the expected relation between these invariants of the transverse push-off of a Legendrian knot and of stabilizations of Legendrian and transverse knots. This leads to one of our main observations, a generalization of Bennequin's inequality.Theorem 2.1 Let (M, ξ) be a tight contact manifold and suppose K is a transverse knot in it of order r > 0 in homology. Further suppose that Σ is a rational Seifert 1

In a lens space X of order r a knot K representing an element of the fundamental group 1 X Š ‫=ޚ‬r ‫ޚ‬ of order s Ä r contains a connected orientable surface S properly embedded in its exterior X N .K/ such that @S intersects the meridian of K minimally s times. Assume S has just one boundary component. Let g be the minimal genus of such surfaces for K , and assume s 4g 1. Then with respect to the genus one Heegaard splitting of X , K has bridge number at most 1. 57M27; 57M25 Statement of resultsAny knot K in a lens space X D L.r; q/, r > 0, is rationally nullhomologous, ie OEK D 0 2 H 1 .X I ‫/ޑ‬ Š 0. We say r is the order of the lens space X , and we say the smallest positive integer s such that sOEK D 0 2 H 1 .X I ‫/ޚ‬ Š ‫=ޚ‬r ‫ޚ‬ is the order of the knot K . Note s Ä r . The exterior X N .K/ of K thus contains a connected properly embedded orientable surface S such that when S is oriented @S is coherently oriented on @ x N .K/ and intersects the meridian  @ x N .K/ of K minimally s times, ie j @S j D s . Such a surface S is an analogue of a Seifert surface for a knot in S 3 . We refer to the genus of a knot K in X as the minimal genus of these "rational" Seifert surfaces for K . For this article we will restrict our attention to knots with rational Seifert surfaces that have just one boundary component.In this paper we prove the following theorem. Theorem 1.1 Let K be a genus g knot of order s in a lens space X whose Seifert surfaces have one boundary component. If s 4g 1 then, with respect to the Heegaard torus of X , K has bridge number at most 1. Theorem 1.1 may be curiously rephrased as saying small genus knots in lens spaces have small bridge number.In [1] Berge shows that double-primitive knots (ie simple closed curves that lie on a genus 2 Heegaard surface in S 3 and represent a generator of 1 for each handlebody)