On left-orderable fundamental groups and Dehn surgeries on twist knots

Abstract: We show that the resulting manifold by $r$-surgery on the hyperbolic twist knot $K_m, \, m \ge 2$, has left-orderable fundamental group if the slope $r$ satisfies the condition $r \in (-4,2m)$ if $m$ is even, and $r \in [0,4] \cup (\frac{4m}{\omega}+4, 2m+4)$ if $m$ is odd, where $\omega>1$ is the unique real solution of the equation $x e^{x}=4m$.