Gopakumar-Vafa invariants of $\mathcal{O}(k)\oplus \mathcal{O}(-k-2)$

We prove a closed formula for leading Gopakumar-Vafa invariants of local Calabi-Yau 3-folds of $\mathcal{O}(k)\oplus \mathcal{O}(-k-2)\rightarrow \mathbb{P}^1$ by the topological vertex. We also obtain some results not predicted before.