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Perturb a clean self-intersection circle of a Lagrangian immersion by a Morse function on $S^1$. Its minimum and maximum give two transverse double points. Resolving both by Lagrangian surgery adds two one-handles and changes the Euler characteristic by $-4$. One neck may be glued with the orientation-reversing choice, so the resolved surface is nonorientable.

If ${\gamma }_i:L_i\to (M_i,{\omega }_i)$ are one-dimensional immersed curves in symplectic surfaces, thenis a Lagrangian immersion. A double point of one factor times the other curve becomes a clean self-intersection circle.