Shadows of 2-knots and complexity

We introduce a new invariant for a $2$-knot in $S^4$, called the shadow-complexity, based on the theory of Turaev shadows, and we give a characterization of $2$-knots with shadow-complexity at most $1$. Specifically, we show that the unknot is the only $2$-knot with shadow-complexity $0$ and that there exist infinitely many $2$-knots with shadow-complexity $1$.