ar X iv : 0 90 1 . 03 80 v 2 [ m at h . SG ] 6 J ul 2 00 9 RATIONAL LINKING AND CONTACT GEOMETRY

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, ThurstonBennequin 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. 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 null-homologous 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 null-homologous 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 slQ(K) of a transverse knot K and the rational Thurston-Bennequin invariant tbQ(L) and rational rotation number rotQ(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 surface of K. Then slQ(K) ≤ − 1 r χ(Σ). Moreover, if K is Legendrian then tbQ(K) + | rotQ(K)| ≤ − 1 r χ(Σ). The first author was partially supported by NSF Grant DMS-0239600. The second author was partially supported by NSF Grants DMS-0239600 and DMS-0804820.