On left-orderability and cyclic branched coverings

TL;DR

This paper explores conditions under which the fundamental groups of cyclic branched coverings of knots are left-orderable, applying recent criteria to classify such properties for a broad class of two-bridge knots.

Contribution

It extends previous work by applying Hu's criterion to a large class of two-bridge knots, identifying specific ranges of r where the coverings are left-orderable.

Findings

Identifies ranges of r for which cyclic branched coverings are left-orderable.

Applies a recent criterion to a broad class of two-bridge knots.

Provides new insights into the relationship between knot properties and group orderability.

Abstract

In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of the integer r>1 for which the r-th cyclic branched covering of S^3 along the knot is left-orderable.