The stable Andrews-Curtis conjecture in combinatorial group theory is the statement that every balanced presentation of the trivial group can be simplified to the trivial form by elementary moves corresponding to "handle-slides" together with "stabilization" moves. Schoenflies conjecture is the statement that the complement of any smooth embedding of S³ into S⁴ is a pair of smooth balls. Here we suggest an approach to these problems by a cork twisting operation on contractible manifolds, and demonstrate it on the example of the first Cappell-Shaneson homotopy sphere.