Left-orderable, non-$L$-space surgeries on knots

Abstract

Let K be a knot in the 3–sphere S3. An r–surgery on K is leftorderable if the resulting 3–manifold K(r) of the surgery has left-orderable fundamental group, and an r–surgery on K is called an L–space surgery if K(r) is an L–space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be classified into left-orderable surgeries or L– space surgeries. We introduce a way to provide knots with left-orderable, non– L–space surgeries. As an application we present infinitely many hyperbolic knots on each of which every nontrivial surgery is a hyperbolic, left-orderable, non–L–space surgery.