Maximal Thurston–bennequin Number of +adequate Links

The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling; therefore its Thurston–Bennequin number is maximal among Legendrian representatives of A. We derive consequences for the Kauffman polynomial and Khovanov homology of +adequate links. The maximum Thurston–Bennequin number, denoted by tb, is a knot invariant that has recently drawn a lot of interest. Its definition is possible because Bennequin’s inequality, tb ≤ 2g − 1, bounds from above the Thurston–Bennequin number of Legendrian representatives of any knot type by (essentially) the genus g of the knot. Either Bennequin’s inequality itself or other bounds, for example the so-called Kauffman bound on tb [10], make the extension to links possible. Recall that the Thurston–Bennequin number is computed from an oriented front diagram by subtracting the number of right cusps from the writhe; tb is the maximum of these numbers for all fronts representing a given link type. The Kauffman bound states that tb, and thus tb, is strictly less than the minimum v–degree (or −1 times the maximum a–degree) of the Kauffman polynomial. The value of tb is known for the following infinite classes of knots and links: positive links [12] (see also [15]), 2-bridge links [8], and more generally, alternating links [9], negative torus knots [3], and Whitehead doubles with sufficiently negative framing [4]. In the positive and alternating cases, the proofs of Tanaka and Ng proceed as follows: for the given knot or link, they construct a certain front diagram. Tanaka uses the Kauffman bound (i.e., establishes that it’s sharp for the front) to show that its Thurston–Bennequin number is maximal. Ng uses the so-called Khovanov bound for the same purpose, but it turns out that the Kauffman bound works just as easily Received by the editors November 9, 2006. 2000 Mathematics Subject Classification. Primary 57M25; Secondary 53D12. 1 For the standard definitions of Legendrian knot theory we refer the reader to [2], [6], or to any number of other publications. Let us only state that we work in Rxyz where the contact structure is the kernel of dz − ydx, so that the front projection is the xz–projection. Similarly, we will not define many other notions of knot theory either, but use [1] as a standard reference instead. 2It seems to be standard to use either v and z or a = v−1 and z as the variables in the Kauffman polynomial. The ‘Dubrovnik version’ [5], [11] has the same degree distribution. 3Independently, Tanaka [13] also reached the same conclusion as Ng using another observation