Coverings of torus knots in $S^2\times S^1$ and universals

Abstract

Let $t_{\alpha,\beta}\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $\alpha$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{\alpha,\beta}$ if $\alpha\geq3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{\alpha,\beta}$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.