Communications in Analysis and Geometry

Volume 22 (2014)

Number 3

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

Pages: 421 – 449



Kimihiko Motegi (Department of Mathematics, Nihon University, Tokyo, Japan)

Masakazu Teragaito (Department of Mathematics and Mathematics Education, Hiroshima University, Higashi-Hiroshima, Japan)


Let $K$ be a knot in the 3-sphere $S^3$. An $r$-surgery on $K$ is left-orderable 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 non-trivial surgery is a hyperbolic, left-orderable, non-$L$-space surgery.

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