Topological computation of the Stokes matrices of the weighted projective line $\mathbb{P}(1,3)$

The localized Fourier-Laplace transform of the Gau{\ss}-Manin system of $f\colon \mathbb{G}_m \to \mathbb{A}^1,\ x \mapsto x + x^{-3}$ is a $\mathcal{D}_{\mathbb{G}_m}$-module, having a regular singularity at $0$ and an irregular one at $\infty$. By mirror symmetry, it is closely related to the quantum connection of the weighted projective line $\mathbb{P}(1,3)$. Following results of A. D'Agnolo, M. Hien, G. Morando and C. Sabbah from 2017, we compute its Stokes multipliers at $\infty$ by purely topological methods. We compare it to the Gram matrix of the Euler-Poincar\'{e} pairing on $D^b(\text{Coh}(\mathbb{P}(1,3)))$.