Left orderability of cyclic branched covers of rational knots $C(2n+1,2m,2)$

TL;DR

This paper analyzes the fundamental groups of cyclic branched covers of certain rational knots by computing their character varieties, revealing conditions for their left orderability based on real points on these varieties.

Contribution

It provides explicit computations of $ ext{SL}_2(bC)$-character varieties for specific rational knots and links the real points to the left orderability of their cyclic branched covers.

Findings

Character varieties computed for $C(2n+1,2m,2)$ knots.

Left orderability determined via real points on the varieties.

Conditions established for the orderability of the fundamental groups.

Abstract

We compute the nonabelian $\mathrm{SL_2}(\mathbb{C})$-character varieties of the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m$ and $n$ are non-zero integers. By studying real points on these varieties, we determine the left orderability of the fundamental groups of the cyclic branched covers of $C(2n+1,2m,2)$.