page 18 : genus $k$ upper bounds twin-width by $\mathcal O(k)$ – The twin-width of every graph $G$ of Euler genus $g \ge 1$ is at most … $18 \sqrt{47g}+O(1)$.
page 8 : genus $k$ upper bounds acyclic chromatic number by $\mathcal O(k)$ – Lemma 4.8 ([3]). The accylic chromatic number $\chi_a$ is upper bounded by the genus $g$. We have $\chi_a \le 4g+4$.
page 10 : feedback edge set $k$ upper bounds genus by $\mathcal O(k)$ – Lemma 4.19. The feedback edge set number $f$ upper bounds the genus $g$. We have $g \le f$.
feedback edge set $k$ upper bounds genus by $\mathcal O(k)$ – Removing $k$ edges creates a forest that is embeddable into the plane. We now add one handle for each of the $k$ edges to get embedding into $k$-handle genus.