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}
}
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… 

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