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

TL;DR

This paper investigates branched coverings of $S^2 imes S^1$ over Seifert manifolds with specific branch sets, computes invariants of Abelian covers, and demonstrates that certain torus knots are not universal in this context.

Contribution

It characterizes when Seifert manifolds are branched covers of $S^2 imes S^1$ with specific branch sets and computes invariants of Abelian covers, also showing non-universality of a particular torus knot.

Findings

Any Seifert manifold with Euler number zero is a branched cover of $S^2 imes S^1$ with branch set $t_{\alpha,\beta}$ for $\alpha\geq 3.

Computed Seifert invariants of Abelian covers of $S^2 \times S^1$ branched along $t_{\alpha,\beta}$.

Proved that $t_{2,1}$ is not a universal knot in $S^2 \times S^1$.

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.