Two smooth manifolds \(X_1\) and \(X_2\) are said to be exotic if they are homeomorphic but not diffeomorphic. If \(Y\) is a codimension \(0\) compact submanifold of a smooth manifold \(X\) and the boundary \(\partial Y\) of \(Y\) is smoothly embedded in \(X\), then for a manifold \(Z\) whose boundary \(\partial Z\) is diffeomorphic to \(\partial Y\), the cut-and-paste \((X-Y)\cup_\varphi Z\) is denoted by \(X(Y,\varphi,Z)\), where \(\varphi:\partial Z\to\partial Y\) is the gluing map. If \(Z=Y\), then the surgery \(X(Y,\varphi)\) is called a twist. If \(C\) is a contractible \(4\)-manifold, \(t:\partial C\to\partial C\) is a self-diffeomorphism on the boundary, and \(t\) cannot extend to a diffeomorphism \(C\to C\), then \((C,t)\) is called a cork. If \((C,t)\) is a cork and \(X\) is a \(4\)-manifold, then for an embedding \(C\hookrightarrow X\) the twist \(X(C,t)\) is called a cork twist. If \(X(C,t)\) is not diffeomorphic to \(X\), then \((C,t)\) is called a cork of \(X\). For a cork \((C,t)\), an integer \(n\) is called the cork order of \((C,t)\) if for any integer \(i\) with \(0<i<n\), the composition \(t\circ t\circ\dots\circ t=t^i\) cannot extend to any diffeomorphism \(C\to C\) and \(t^n\) can extend to a diffeomorphism \(C\to C\). If the cork order of \((C,t)\) is \(n\), then we call the cork an \(n\)-cork.
For a contractible \(4\)-manifold \(C\), the goal of this paper is to construct infinite families of \(n\)-corks and give a technique to show that the map \(t\) for a twist \((C,t)\) cannot extend to the inside \(C\) as a diffeomorphism. In [Invent. Math. 123, No. 2, 343–348 (1996; Zbl 0843.57020)], C. L. Curtis et al. showed that for any simply-connected closed exotic \(4\)-manifolds \(X_1\) and \(X_2\), there exist a contractible Stein \(4\)-manifold \(C\), an embedding \(C\hookrightarrow X_1\), and a self-diffeomorphism \(t:\partial C\to\partial C\) with \(t^2=\text{id}\) such that \(X_1(C,t)=X_2\). In this paper, the author shows that for \(n>1\) and \(m>0\) there exists a cork \((C_{n,m},\tau_{n,m}^C)\) with cork order \(n\), and \(C_{n,m}\) is a Stein cork with cork order \(n\). Also, it is shown that for \(n>1\) and \(m>0\) there exists an embedding \(D_{n,m}\hookrightarrow D_{1,m}=C(m)\) such that the twist \(C(m)(D_{n,m},\tau_{n,m}^D)\) for the embedding gives a diffeomorphism \(\psi:C(m)\cong C(m)(D_{n,m},\tau_{n,m}^D)\) and the original identity map \(\text{id}|_{\partial D_{1,m}}\) induces \(\tau(m)\) with respect to the restriction of the diffeomorphism \(\psi|_{\partial C(m)}\). Hence, \((D_{n,m},\tau_{n,m}^D)\) is an \(n\)-cork.