On the Additivity of the Thurston-bennequin Invariant of Legendrian Knots

The standard contact structure ξ0 on 3-space R3 = {(x, y, z)} is the plane field on R3 given by the kernel of the 1-form dz−ydx. A Legendrian knot K in the contact manifold (R3, ξ0) is a knot which is everywhere tangent to the contact structure ξ0. The Thurston-Bennequin invariant tb(K) of a Legendrian knot K in (R3, ξ0) is the linking number of K and a knot K ′ which is obtained by moving K slightly along the vector field ∂ ∂z . For a topological knot type k in R3, the maximal Thurston-Bennequin invariant mtb(k) is defined to be the maximal value of tb(K), where K is a Legendrian knot which topologically represents k. For any k, by the Bennequin’s inequality in [1], we know that mtb(k) is an integer (i.e., not ∞). There are several computations of mtb(k) (for example, see [3], [5], [8], [9], [10], [11]). In this paper, we prove the following theorem: