Links all of whose cyclic branched covers are L-spaces

TL;DR

This paper proves that certain pretzel knots have all their cyclic branched covers as L-spaces, and explores their properties, including quasipositivity and sliceness, extending known results to new families of knots and links.

Contribution

It establishes that specific pretzel knots have all cyclic branched covers as L-spaces and demonstrates their quasipositivity and sliceness, extending previous classifications to new families.

Findings

All cyclic branched covers of the pretzel knots K_k are L-spaces.

Knots K_k with k≥1 are quasipositive and slice.

Extended results to two-bridge links with all L-space cyclic branched covers.

Abstract

We show that for the pretzel knots $K_k=P(3,-3,-2k-1)$, the $n$-fold cyclic branched covers are L-spaces for all $n\geq 1$. In addition, we show that the knots $K_k$ with $k\geq 1$ are quasipositive and slice, answering a question of Boileau-Boyer-Gordon. We also extend results of Teragaito giving examples of two-bridge knots with all L-space cyclic branched covers to a family of two-bridge links.