Legendrian and Transversal Knots

Linear Legendrian curves in $T^3$

“…The dividing set of A consists of a single dividing arc with both endpoints on T 2 × {0}, and the simply connected region of A \ Γ A is positive (negative). Then by [3], Lemma 2.20, a vertical Legendrian ruling of T 2 × {0} is a positive (negative) stabilisation of {p 1 } × S 1 . From the well definedness of stabilisation, it follows that L 1 is contact isotopic to a vertical Legendrian ruling of T 2 × {0}.…”

“…Contrarily to stabilisations, destabilisations are not always possible. In fact a destabilisation is equivalent to finding a bypass: see [3], Lemma 2.20.…”

“…Stabilisation is performed in the standard R 3 by adding a zig-zag to the front projection of the Legendrian knot, and the definition can be extended to Legendrian knots in any contact manifold by Darboux Theorem. See for example [3], Section 2.7, or [5], Section 2.1. A stabilisation comes with a sign, which corresponds to the choice of one of the two possible ways to add a zig-zag to the front projection.…”

Ozsváth-Szabó invariants and fillability of contact structures

“…Let S + and S − denote the operations of positive and negative stabilisation defined, for example, in [4], Section 2.7. Given i ∈ P * n , denote the contact structure on −Σ(2, 3, 6n+5) obtained by Legendrian surgery on (M 0 , ξ 1 ) along the Legendrian knot S (n−1+i)/2 + S (n−1−i)/2 − (F ) by η i .…”

Rational symplectic field theory for Legendrian knots

“…The theory of Legendrian knots plays a key role in contact and symplectic topology and has recently shown surprising connections to low dimensional topology; see [Etn05] for a survey of the subject. A key breakthrough in the study of Legendrian knots, and symplectic topology generally, was the introduction of Gromov-type holomorphic-curve techniques in the 1990s.…”