Twist left-veering open books and overtwisted contact structures

We introduce a notion of $N$-twist left-veering mapping classes and prove that an open book with $N$-twist left-veering monodromy contains an overtwisted disk that intersects the binding at $N$ points. As an application we show that an open book is $2$-twist (resp. $1$-twist) left-veering if and only if the depth of the binding is $2$ (resp. $1$).