Powers of Dehn twists generating right-angled Artin groups

We give a bound for the exponents of powers Dehn twists to generate right-angled Artin group. Precisely, if $\mathcal{F}$ is finite collection pairwise distinct simple closed curves on type surface and $N$ denotes maximum intersection numbers all pairs in $\mathcal{F}$, then we prove that $\{T_\gamma^n \,\vert\, \gamma \in \mathcal{F} \}$ generates group $n \geq N^2 + N 3$. This extends previous result Koberda, who proved existence possibly depending underlying hyperbolic structure surface. In course proof, obtain universal only topological certain cases, which partially answers question due Koberda.