On cabled knots, Dehn surgery, and left-orderable fundamental groups

TL;DR

This paper refines a criterion for when Dehn surgery on knots yields non-left-orderable fundamental groups, introducing the concept of decayed knots and exploring their properties and implications in knot theory.

Contribution

It introduces the notion of decayed knots, refines the existing criterion for non-left-orderability, and proves that positive cables of decayed knots are also decayed, linking to L-space surgery properties.

Findings

Decayed knots produce non-left-orderable groups after sufficiently positive surgeries.

Positive cables of decayed knots are themselves decayed.

Results mirror properties of L-space surgeries in Heegaard Floer homology.

Abstract

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of L-space surgeries in the context of Heegaard Floer homology.