An Elementary Approach on Left-Orderability, Cables of Torus Knots and Dehn Surgery

TL;DR

This paper provides an elementary proof that certain Dehn surgeries on cable knots of torus knots produce 3-manifolds with non-left-orderable fundamental groups, addressing a specific question in knot theory.

Contribution

It offers a new elementary approach to determine non-left-orderability of fundamental groups after surgeries on cable knots of torus knots, extending previous results.

Findings

Surgeries with slope in [pq-1,pq] on specific cable knots yield non-left-orderable groups.

Elementary methods can be used to analyze left-orderability in this context.

Results partially answer Clay and Watson's question on left-orderability of these surgeries.

Abstract

Motivated by Clay and Watson's question on left-orderability of the fundamental group of the resultant space of an $r'$-surgery on the $(p, q)$-cable knots for $r' \in (pq-p-q,pq)$, this paper proves by elementary means that for specific pairs of $(p,q)$-cable knots of torus knots, $r' \in [pq-1,pq]$ gives a surgery yielding non-left orderable fundamental groups.