Left-orderability and exceptional Dehn surgery on twist knots (Q2862965)

For \(|r|\leq 4\), \(r\)-surgery on the figure eight knot yields a \(3\)-manifold with left-orderable fundamental group. This was shown by \textit{S. Boyer} et al. [Math. Ann. 356, No. 4, 1213--1245 (2013; Zbl 1279.57008)] for \(|r|<4\), and \textit{A. Clay} et al. [Algebr. Geom. Topol. 13, No. 4, 2347--2368 (2013; Zbl 1301.06038)] for \(r=|4|\).NEWLINENEWLINEIn this paper the author shows that \(4\)-surgery on the \(m\)-twist knot with \(|m|\geq 2\) yields a manifold \(M\) with left-orderable fundamental group.NEWLINENEWLINEHe first shows that \(4\)-surgery for \(m\neq 0,-1\) yields a graph manifold \(M\) that is a union of the knot exterior \(M_1\) of the torus knot \((2,2m+1)\) and the twisted \(I\)-bundle \(M_2\) over the Klein bottle. Thus the fundamental group of \(M\) is a free product of \(\pi_1 (M_1 )\) and \(\pi_1 (M_2 )\) with amalgamation along \(\pi_1 (\partial M_i )\). He then exhibits a normal family \(L_i\) of left-orderings of \(\pi_1 (M_i )\) (\(i=1,2\)) and shows that these induce a left-ordering of \(\pi_1 (M )\).