Zeeman theorem on twist-spun knots

Let $K\subset S^3$ be a smooth knot and let $m\ne 0$ be an integer. The exterior of its $m$-twist-spun knot in $S^4$ is a fiber bundle over a circle, with fiber the $|m|$-fold cyclic branched covering of $S^3$ along $K$ with an open three-ball removed. For the open complement, use the corresponding branched cover with a point removed. Its monodromy comes from the cyclic deck action.^[ref]

For $m=\pm 1$, the branched cover is $S^3$, the compact fiber is a three-ball, and the resulting two-dimensional surface knot is unknotted. Zero-twist spinning is ordinary spinning and is excluded from this nonzero-twist fibration theorem; a zero-fold branched cover is not its fiber.^[ref]