Differential algebra of Legendrian links

Abstract

Let the space R = {(q, p, u)} be equipped with the standard contact form α = du − pdq. A link L ⊂ R3 is called Legendrian if the restriction of α to L vanishes. Two Legendrian links are said to be Legendrian isotopic if they can be connected by a path in the space of Legendrian embeddings. Consider the map π : R3 → R2, (q, p, u) → (q, p). Since the kernel of dπ is everywhere transverse to the kernel of α, the projection π(L) of a Legendrian link L is an immersed curve. We say that a link L ⊂ R3 is generic with respect toπ, or π-generic, if all self-intersections of π(L) are transverse double points. Given a π-generic Legendrian link L , its diagram is the curve π(L) ⊂ R2, at every crossing of which the overpassing branch (the one with the larger value of u) is marked. It is well known that every smooth link type can be realized by a Legendrian link (it is shown in Sect. 11 that such a realization can be chosen to have a diagram with relatively few crossings). The so-called classical invariants of an oriented Legendrian link L are defined as follows. First of them is just the smooth isotopy type of L . Let L1, . . . , Lk be the components of L . The Maslov number m(Li) of Li is twice the rotation number of π(Li). The Thurston–Bennequin number β(Li) of Li is the linking number between Li and s(Li), where s is a small shift along the u direction. The orientation on R3 involved in the definition of the linking number is given by the form α ∧ dα = dq ∧ dp ∧ du. In terms of the diagram, β(Li) is the number of double points of π(Li) counted with signs shown in Fig. 1. The Thurston–Bennequin number does not depend on the orientation of Li , while the Maslov number changes its sign when the orientation changes.