Let $\mathfrak{D}\subset B$ be a graph with one vertex emitting 3 edges. Topologically, we can present $\mathfrak{D}$ as

(1.2)

$$\begin{split}\mathfrak{D}&=\mathfrak{D}_{1}\cup\mathfrak{D}_{2}\cup\mathfrak{D}_{3}\cup\{0\}\\ &=\{\mu_{1}=0,\mu_{2}>0\}\cup\{\mu_{2}=0,\mu_{1}>0\}\cup\{\mu_{1}=\mu_{2}<0\}\cup\{0\}\\ &\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\{0\}\subset\mathbb{R}^{2}\times\mathbb{R}=B.\end{split}$$

The total space $M^{+}$ is built as a singular $T^{2}$-bundle over $B\times S^{1}=\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}\times S^{1}$ with discriminant locus $\mathfrak{D}\times\{0\}\subset B\times S^{1}$. Let $e_{1},e_{2}$ denote a basis of $H_{1}(T^{2},\mathbb{Z})$, and denote $T(ae_{1}+be_{2})$ as the subtorus with homology class $ae_{1}+be_{2}$. Over $(B\times S^{1})\setminus(\mathfrak{D}\times\{0\})$ the space $M^{+}$ is a principal $T^{2}$-bundle, whose Chern class $c_{1}\in H^{2}(B\times S^{1}\setminus(\mathfrak{D}\times\{0\}),\mathbb{Z}e_{1}\oplus\mathbb{Z}e_{2})$ evaluates to $e_{1},-e_{2},-e_{1}+e_{2}$ respectively on the $S^{2}$-cycles linking $\mathfrak{D}_{1}\times\{0\},\mathfrak{D}_{2}\times\{0\},\mathfrak{D}_{3}\times\{0\}$ inside $B\times S^{1}$. Over the codimension 3 loci $\mathfrak{D}_{1}\times\{0\}$, $\mathfrak{D}_{2}\times\{0\}$, $\mathfrak{D}_{3}\times\{0\}$ inside $B\times S^{1}$, the $T^{2}$-fibres collapse to circle fibres $T^{2}/T(e_{1})$, $T^{2}/T(-e_{2})$, $T^{2}/T(-e_{1}+e_{2})$ respectively. Finally, over the origin $\{0\}\subset B\times S^{1}$, the $T^{2}$-fibre collapses to a point. The singular $T^{2}$-bundle over a small neighbourhood of the origin $D^{4}\subset B\times S^{1}$ is topologically modelled on

(1.3)

$$\begin{split}&\pi_{\mathbb{C}^{3}}:\mathbb{C}^{3}\to\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}\times\mathbb{R},\\ &(z_{0},z_{1},z_{2})\mapsto\left(\frac{1}{2}(|z_{1}|^{2}-|z_{0}|^{2}),\frac{1}{2}(|z_{2}|^{2}-|z_{0}|^{2}),\text{Im}(z_{0}z_{1}z_{2}),\text{Re}(z_{0}z_{1}z_{2})\right)\end{split}$$

whose discriminant locus is compatible with $\mathfrak{D}$.

By construction $M^{+}$ fibres over $B$ with generic fibre $T^{3}$. The singular fibre over $0\in B$ has the topology of $T^{3}$ with $T^{2}$ collapsed to a point, so has Betti numbers $(b_{1},b_{2})=(1,2)$ and Euler characteristic $+1$ (hence the name ‘positive vertex’). A basis of $H_{1}(T^{3})$ is given by $e_{1},e_{2}\in H_{1}(T^{2})\subset H_{1}(T^{3})$ and an $S^{1}$-cycle $e_{0}$ on the total space lifting the cycle $S^{1}\subset B\times S^{1}$. The monodromies around the 3 edges $\mathfrak{D}_{1}$, $\mathfrak{D}_{2}$, $\mathfrak{D}_{3}$ acting on $H_{1}(T^{3})$ are given in the basis $e_{0},e_{1},e_{2}$ as

$$\begin{bmatrix}1&0&0\\ 1&1&0\\ 0&0&1\end{bmatrix},\quad\begin{bmatrix}1&0&0\\ 0&1&0\\ -1&0&1\end{bmatrix},\text{ and }\begin{bmatrix}1&0&0\\ -1&1&0\\ 1&0&1\end{bmatrix}.$$