Cyclic branched covers of Seifert links and properties related to the A D E $ADE$ link conjecture

Abstract

In this article, we show that all cyclic branched covers of a Seifert link have left-orderable fundamental groups, and therefore admit co-oriented taut foliations and are not $L$-spaces, if and only if it is not an $ADE$ link up to orientation. This leads to a proof of the $ADE$ link conjecture for Seifert links. When $L$ is an $ADE$ link up to orientation, we determine which of its canonical $n$-fold cyclic branched covers ${\Sigma }_n\left(L\right)$ have nonleft-orderable fundamental groups. In addition, we give a topological proof of Ishikawa's classification of strongly quasi-positive Seifert links and we determine the Seifert links that are definite, resp., have genus zero, resp. have genus equal to its smooth 4-ball genus, among others. In the last section, we provide a comprehensive survey of the current knowledge and results concerning the $ADE$ link conjecture.