Non-left-orderable surgeries on twisted torus knots

TL;DR

This paper develops a new criterion to determine when Dehn surgeries on twisted torus knots produce 3-manifolds with non-left-orderable fundamental groups, supporting the Boyer-Gordon-Watson conjecture.

Contribution

It introduces a generalized criterion for non-left-orderability of quotient groups from Dehn surgeries, extending previous results to a broader class of twisted torus knots.

Findings

Verified the conjecture for a wider class of L-space twisted torus knots.

Provided a new criterion for non-left-orderability of quotient groups.

Extended previous theorems to more general cases.

Abstract

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots in the 3-sphere, it is natural to ask what conditions on the knot group are sufficient to imply that the quotient associated to Dehn surgery is not left-orderable. Clay and Watson develop a criterion for determining the left-orderability of this quotient group and use it to verify the conjecture for surgeries on certain L-space twisted torus knots. We generalize a recent theorem of Ichihara and Temma to provide another such criterion. We then use this new criterion to generalize the results of Clay and Watson and to verify the conjecture for a much broader class of L-space twisted torus knots.