Capovilla-Searle, O; Traynor, L
Source: Pacific Journal of Mathematics, 285 (2):319-343; 10.2140/pjm.2016.285.319 DEC 2016
In the symplectization of standard contact 3-space, R x R-3, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus 0. We show that any Legendrian knot has a nonorientable Lagrangian endocobordism, and that the cross-cap genus of such a nonorientable Lagrangian endocobordism must be a positive multiple of 4. The more restrictive exact, nonorientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do exist for any stabilized Legendrian knot. Moreover, the relation defined by exact, nonorientable Lagrangian cobordism on the set of stabilized Legendrian knots is symmetric and defines an equivalence relation, a contrast to the nonsymmetric relation defined by orientable Lagrangian cobordisms.