The fundamental ideal and $\pi _{2}$ of higher dimensional knots

Abstract:

Let $({S^4},k({S^2}))$ be a knot formed by spinning a polyhedral arc $\alpha$ about the standard $2$-sphere ${S^2}$ in the $3$-sphere ${S^3}$. Then the second homotopy group of ${S^4} - k({S^2})$ as a $Z{\pi _1}$-module is isomorphic to each of the following: (1) The fundamental ideal modulo the left ideal generated by $a - 1$, where $a$ is the image in ${\pi _1}({S^4} - k({S^2}))$ of a generator of ${\pi _1}({S^2} - \alpha )$. (2) The first homology group of the kernel of ${\pi _1}({S^3} - k({S^2})) \to {\pi _1}({S^4} - k({S^2}))$