Clean-circle resolution of a Lagrangian surface immersion

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.

For every $l\ge 1$, take a plane immersed circle with $l$ transverse double points and form its product with an embedded circle in a second symplectic plane. The resulting torus is a Product Lagrangian immersion with $l$ clean double circles. Resolving them gives a connected nonorientable Lagrangian embedding in ${\mathbb{R}}^4$ with Euler characteristic $-4l$.