Chekanov-eliashberg Invariants and Transverse Approximations of Legendrian Knots

In this article, we consider Legendrian and transverse knots in the standard contact space, that is in R3 with the contact structure globally given by the 1-form α = ydx− dz. It is well-known that a little push of an oriented Legendrian knot Γ in the direction of its positive normal within the contact structure changes it into a transverse knot Γ+, whose natural orientation, as determined by α, matches the chosen orientation of Γ; pushing Γ in the direction of its negative normal produces the transverse knot Γ− of the opposite orientation. Any transverse knot is transverse isotopic to Γ+ derived from some oriented Legendrian knot Γ, and if Legendrian knots Γ,Γ′ are Legendrian isotopic, then Γ+,Γ′+ are transverse isotopic. Besides their topological invariants, Legendrian and transverse knots have classical invariants of contact origin: The Thurston-Bennequin number τβ(Γ) and the Maslov number μ(Γ) for a Legendrian knot Γ and the ThurstonBennequin number τβ(Γ) for a transverse knot Γ. Furthermore, τβ(Γ) = τβ(Γ) + μ(Γ), τβ(Γ−) = τβ(Γ)− μ(Γ).