Universal polynomials for Gromov-Witten invariants on toric surfaces

We study the polynomiality of combined logarithmic and ordinary Gromov-Witten invariants with a Lambda class insertion on toric surfaces $Δ(\mathbf{c},\mathbf{d})$. We show that the combined logarithmic Gromov-Witten invariants can be expressed in terms of universal relative node polynomials when the direction vector $\mathbf{c}$ and length vector $\mathbf{d}$ of the $h$-transverse lattice polygon $Δ(\mathbf{c},\mathbf{d})$ are sufficiently large and spread out. In particular, when the normal fan of the $h$-transverse lattice polygon $Δ(\mathbf{c},\mathbf{d})$ is smooth and its edges satisfy a length condition, we show that the combined Gromov-Witten invariants are equal to a universal node polynomial. We also show that the higher genus Block-Göttsche polynomials of smooth toric surfaces corresponding to these $h$-transverse lattice polygons, including toric blowups beyond Hirzebruch and toric del Pezzo surfaces, are equal to a generating function of higher genus Gromov-Witten invariants.