The Kirby-calculus reformulation of the smooth four-dimensional Poincaré conjecture

Let $L=L_1\cup L_2$ be a link in $S^3$, with $L_1$ a dotted $p$-component unlink and $L_2$ a framed link of $p+q$ components. Suppose that $L_2$ normally generates $\pi_1(S^3-L_1)$, and that surgery on $L$, with dotted components 0-framed, is diffeomorphic to $S_q$. Kirby-calculus reformulation of SPC4. (a) There is a sequence of moves (1), (2), and (3) transforming $L$ to the empty diagram. (b) If all framings on $L_2$ are even, the sequence can be chosen with $n=0$ in every move (1). This is a link-diagram formulation of SPC4, expressing the absence of exotic smooth homotopy 4-spheres through Kirby moves.