Slope of Orderable Dehn Filling of Two-Bridge Knots

TL;DR

This paper investigates the relationship between rational slopes in Dehn fillings of two-bridge knots and the left-orderability of their fundamental groups, providing computational results and conjectures.

Contribution

It introduces a method to compute the range of slopes yielding left-orderable fundamental groups and proposes a conjecture for two-bridge knots.

Findings

Computed the slope range for left-orderability in double twist knots

Identified patterns in Riley polynomial roots related to left-orderability

Proposed a conjecture on surgery slopes for two-bridge knots

Abstract

In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope $r$ such that $r$-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots.