• DOI:10.2140/pjm.2016.285.319
• Corpus ID: 119315987

Nonorientable Lagrangian cobordisms between Legendrian knots

@article{CapovillaSearle2015NonorientableLC,
  title={Nonorientable Lagrangian cobordisms between Legendrian knots},
  author={Orsola Capovilla-Searle and Lisa Traynor},
  journal={Pacific Journal of Mathematics},
  year={2015},
  volume={285},
  pages={319-343}
}

• Orsola Capovilla-Searle, L. Traynor
• Published 11 August 2015
• Mathematics
• Pacific Journal of Mathematics

In the symplectization of standard contact $3$-space, $\mathbb R \times \mathbb R^3$, 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 non-orientable Lagrangian endocobordism, and that the crosscap genus of such a non-orientable Lagrangian endocobordism must be a positive multiple of $4$. The more restrictive exact, non-orientable 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, non-orientable Lagrangian cobordism on the set of stabilized Legendrian knots is symmetric and defines an equivalence relation, a contrast to the non-symmetric relation defined by orientable Lagrangian cobordisms.