We construct a tropical Lagrangian torus $L_{T^2}\subset \mathbb{CP}^2\setminus E$ which is not isotopic to the standard product torus $F_q$ in $\mathbb{CP}^2\setminus E$. We then investigate a symplectomorphism of $\mathbb{CP}^2\setminus E$ which interchanges $F_q$ and $L_{T^2}$. We speculate on the action of this symplectomorphism on the Fukaya category and its relation to homological mirror symmetry.

This work was partially supported by NSF grants DMS-1406274 and DMS-1344991 and by a Simons Foundation grant (#385573, Simons Collaboration on Homological Mirror Symmetry).