Define a special Lagrangian $T^{2}$-cone $C$ in ${\mathbin{\mathbb{C}}}^{3}$ by

$$C=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:|z_{1}|=|z_{2}|=|z_{3}|,\;\>z_{1}z_{2}z_{3}\in[0,\infty)\bigr\}.$$

(2.4)

This will be important in §3.6 as it is a ‘stable’ special Lagrangian singularity in the sense of [33, Def. 3.6]. There are three families of explicit asymptotically conical SL 3-folds $L^{A}_{1},L^{A}_{2},L^{A}_{3}$ for $A>0$ in ${\mathbin{\mathbb{C}}}^{3},$ each diffeomorphic to ${\mathbin{\cal S}}^{1}\times{\mathbin{\mathbb{R}}}^{2}$ and asymptotic at rate $\rho=0$ to the cone $C$, where

$$L^{A}_{1}=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:|z_{1}|^{2}-A=|z_{2}|^{2}=|z_{3}|^{2},\;\>z_{1}z_{2}z_{3}\in[0,\infty)\bigr\},$$

(2.5)

and $L^{A}_{2},L^{A}_{3}$ are obtained from $L^{A}_{1}$ by cyclic permutation of $z_{1},z_{2},z_{3}$.