On Hyperkähler manifolds of K3${}^{[n]}$-type with large Picard number

Inspired by well-known examples of hyperkähler manifolds, we show that any hyperkähler manifold $X$ of K3${}^{[n]}$-type with Picard number $\rho (X)\ge 4$ is always isomorphic to a moduli space of twisted stable sheaves on a K3 surface. Additionally, we provide explicit descriptions of hyperkähler manifolds of K3${}^{[n]}$-type with Picard ranks below this crucial value (e.g., $\rho (X)=3$) that are not birational to such moduli spaces.