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A cork is a smooth, contractible, oriented, compact 4-manifold $W$ togetherwith a self-diffeomorphism $f$ of the boundary 3-manifold that cannot extend toa self-diffeomorphism of $W$; the cork is said to be strong if $f$ cannotextend to a self-diffeomorphism of any smooth integer homology ball bounded by$\partial W$. Surprising recent work of Dai, Hedden, and Mallick showed thatmost of the well-known corks in the literature are strong. We construct thefirst non-strong corks, which also give rise to new examples of absolutelyexotic Mazur manifolds. Additionally we give the first examples of corks wherethe diffeomorphism of $\partial W$ can be taken to be orientation-reversing.

New curiosities in the menagerie of corks