In the most symmetric case $\phi_{1}=\phi_{2}=\ldots\phi_{n}=\frac{\pi}{n}$, the Lawlor neck is invariant under $SO(n,\mathbb{R})$, and these holomorphic triangles are up to $SO(n,\mathbb{R})$ rotation, simply the triangle inside the first coordinate line $\mathbb{C}\subset\mathbb{C}^{n}$ enclosed by the three Lagrangians.