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By a result due to Ziltener, there exist no closed embedded Bohr-SommerfeldLagrangians inside $\mathbb CP^n$ for the prequantisation bundle whose totalspace is the standard contact sphere. On the other hand, any embedded monotoneLagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeldimmersion. We draw the front projections for the corresponding Legendrian liftsinside a contact Darboux ball of the threefold covers of both thetwo-dimensional Clifford and Chekanov tori (the former is the Legendrian linkof the Harvey-Lawson special Lagrangian cone), and compute the associatedChekanov-Eliashberg algebras. Although these Legendrians are not loose, we showthat they both admit exact Lagrangian cobordisms to the loose Legendriansphere; they hence admit exact Lagrangian caps in the symplectisation, whichare non-regular Lagrangian cobordisms. Along the way, we also computebilinearised Legendrian contact homology of a general Legendrian surface in thestandard contact vector space when all Reeb chords are of positive degree, aswell as the augmentation variety in the case of tori.

Legendrian submanifolds from Bohr-Sommerfeld covers of monotone Lagrangian tori