We recount here the historical perspective of Gross and Ruan on negative vertices, to be modified in Section 1.1.5. Let $\mathfrak{D}\subset B$ be a graph with one vertex emitting 3 edges. Topologically, we present $\mathfrak{D}$ as

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

Let $e_{1},e_{2}$ be a basis of $H_{1}(T^{2},\mathbb{Z})$. Let $S\subset B\times T^{2}$ be a ‘pair of pants’ (namely a surface homeomorphic to the complement of 3 points in $S^{2}$) sitting over $\mathfrak{D}\subset B$, such that $S\cap(\mathfrak{D}_{i}\times T^{2})$ is a cylinder $\mathfrak{D}_{i}\times S^{1}$, where the $S^{1}$ factor inside $T^{2}$ has homology class $e_{2},e_{1},-e_{1}-e_{2}$ for $i=1,2,3$ respectively. The fact that these 3 classes add up to zero means the 3 cylinders $\mathfrak{D}_{i}\times S^{1}$ can be joined together over $\{0\}\times T^{2}$. The fibre of $S\to\mathfrak{D}$ over $0\in\mathfrak{D}$ is a ‘figure 8 diagram’.

Then the total space $M^{-}$ is built as a singular $S^{1}$-bundle over $B\times T^{2}$, which restricts to a principal $S^{1}$-bundle over the complement of the codimension 3 locus $S\subset B\times T^{2}$, and along $S$ the $S^{1}$-fibres collapse to points. The first Chern class of the $S^{1}$-bundle evaluates trivially on $T^{2}\subset B\times T^{2}$ but nontrivially on the $S^{2}$-cycle wrapping $S$. In the 3 transverse directions, the $S^{1}$ fibration is modelled topologically on (1.1).

By construction $M^{-}$ fibres over $B$ with generic fibre $T^{3}$, where $T^{3}$ itself is an $S^{1}$-bundle over $T^{2}$. The class of this $S^{1}\subset T^{3}$ is denoted $e_{3}$. The singular fibre of $M^{-}\to B$ over $0\in B$ is obtained by taking the bundle $T^{3}\to T^{2}$, and collapse down its $S^{1}$-fibres over a ‘figure 8 diagram’ inside $T^{2}$. This singular fibre has Betti numbers $(b_{1},b_{2})=(2,1)$ and Euler characteristic $-1$ (hence the name ‘negative vertex’). The homology classes $e_{1},e_{2}$ lift to $H_{1}(T^{3})$. The monodromies around the edges $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ acting on $H_{1}(T^{3},\mathbb{Z})$ are given in the basis $e_{1},e_{2},e_{3}$ of $H_{1}(T^{3},\mathbb{Z})$ as

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