Generalized square knots and homotopy $4$-spheres

The purpose of this paper is to study geometrically simply-connected homotopy $4$-spheres by analyzing $n$-component links in $S^3$ with a Dehn surgery realizing $\#^n (S^1 \times S^2)$. We call such $n$R-links. Our main result that $4$-sphere can be built without $1$-handles and only two $2$-handles diffeomorphic the standard special case one attached along knot form $Q_{p,q} = T_{p,q} \# T_{-p,q}$, which we generalized square knot. This theorem subsumes prior results Akbulut Gompf. Along way, use thin position techniques from Heegaard theory give characterization $2$R-links component fibered knot, showing second converted via trivial handle additions handleslides derivative link contained fiber surface. invoke Casson Gordon Equivariant Loop Theorem classify handlebody-extensions for closed monodromy $Q_{p,q}$. As consequence, produce large families, all even $n$, $n$R-links are potential counter-examples Generalized Property R Conjecture. also obtain related classification statements fibered, homotopy-ribbon disks bounded knots.