Alternating links and left-orderability

TL;DR

This paper proves that the fundamental group of the branched double-cover of an alternating link in S^3 is left-orderable if and only if the link is an unlink, providing a concise proof of a known result.

Contribution

The paper offers a short, simplified proof of a theorem relating left-orderability of the fundamental group to the unlink status of alternating links.

Findings

Fundamental group of Sigma(L) is left-orderable iff L is an unlink.

Provides a concise proof of a known theorem.

Clarifies the relationship between link properties and group orderability.

Abstract

Let L \subset S^3 denote an alternating link and Sigma(L) its branched double-cover. We give a short proof of the fact that the fundamental group of Sigma(L) admits a left-ordering iff L is an unlink. This result is originally due to Boyer-Gordon-Watson.