Genus one free energy contribution to the quartic Kontsevich model

We prove a formula for the genus one free energy $\mathcal{F}^{(1)}$ of the quartic Kontsevich model for arbitrary ramification by working out a boundary creation operator for blobbed topological recursion. We thus investigate the differences in $\mathcal{F}^{(1)}$ compared with its generic representation for ordinary topological recursion. In particular, we clarify the role of the Bergman $\tau$-function in blobbed topological recursion. As a by-product, we show that considering the holomorphic additions contributing to $\omega_{g,1}$ or not gives a distinction between the enumeration of bipartite and non-bipartite quadrangulations of a genus-$g$ surface.