Mathematical Research Letters
Volume 18 (2011)
On cabled knots, Dehn surgery, and left-orderable fundamental groups
Pages: 1085 – 1095
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.