\tilde{\omega}_i \right\rbrace ,  \tilde{J}_i)$ be the set of $\omega_i$-compatible almost-complex structures on $M_i$ coinciding with $\tilde{J}_i$ outside $\mathrm{int}\left\lbrace \omega_i = \tilde{\omega}_i \right\rbrace$. Let also $$\mathcal{J}_t(M_i, \mathrm{int}\left\lbrace \omega_i = \tilde{\omega}_i \right\rbrace ,  \tilde{J}_i) = \mathcal{C}^\infty([0,1], \mathcal{J}(M_i, \mathrm{int}\left\lbrace \omega_i = \tilde{\omega}_i \right\rbrace ,  \tilde{J}_i))$$ be the set of time-dependent almost-complex structures.

In order to have transverse intersections, we introduce Hamiltonian perturbations. Let $\underline{H} = (H_i)_{i = 1\cdots k}$ be Hamiltonians, $H_i: M_i\times {\mathbb{R}}\rightarrow {\mathbb{R}}$ with support inside $\mathrm{int}\left\lbrace \omega_i = \tilde{\omega}_i \right\rbrace$. Denote $\varphi_i$ the time 1 flow of $X_{H_i}$ and $$\begin{aligned}
& \widetilde{L}_{i(i+1)} = \left\lbrace (\varphi_i(x_i),x_{i+1}) ~|~ (x_i,x_{i+1}) \in L_{i(i+1)} \right\rbrace   \\
 & \widetilde{L}_{(0)} = \widetilde{L}_{0} \times \widetilde{L}_{12} \times \cdots \\
& \widetilde{L}_{(1)} = \widetilde{L}_{01} \times \widetilde{L}_{23} \times \cdots.\end{aligned}$$

Suppose that the generalized intersection points ${\mathcal{I}}(\underline{L})$ are contained inside the product of the $\mathrm{int}\left\lbrace \omega_i = \tilde{\omega}_i \right\rbrace$, which will be the case when defining HSI homology. For a generic choice of Hamiltonians, the intersection $\widetilde{L}_{(0)} \cap \widetilde{L}_{(1)}$ is transverse. The finite set $\widetilde{L}_{(0)} \cap \widetilde{L}_{(1)}$ is in one-to-one correspondence with the set of perturbed generalized intersection points ${\mathcal{I}}_{\underline{H}}(\underline{L})$ consisting of $k$-tuples $p_i\colon [0,1] \rightarrow M_i$ such that $\frac{\mathrm{d}p_i}{\mathrm{d}t} = X_{H_i}$, and $(p_i(1), p_{i+1}(0) )\in L_{i(i+1)}$. Indeed, given $p_i(1) = \varphi_i(p_i(0))$, 