Twin-width can be exponential in treewidth

For any small positive real $\epsilon$ and integer $t > \frac{1}{\varepsilon}$,webuildagraphwithavertexdeletionsetofsize$t$toatree,andtwin-widthgreaterthan$2^{(1-\varepsilon) t}$. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least $2-\epsilon$. Except for the one in oriented twin-width, these lower bounds are essentially tight.