Left-orderable, non-\(L\)-space surgeries on knots (Q402733)

In this paper the authors develop a method to construct knots in the 3-sphere admitting Dehn surgeries yielding 3-manifolds with left-orderable fundamental groups (called left-orderable surgeries), but not admitting Dehn surgeries yielding \(L\)-spaces (called \(L\)-space surgeries). This research is related to the famous \(L\)-space conjecture due to Boyer, Gordon, and Watson raised in [\textit{S. Boyer} et al., Math. Ann. 356, No. 4, 1213--1245 (2013; Zbl 1279.57008)]. The conjecture says that an irreducible rational homology 3-sphere is an \(L\)-space if and only if its fundamental group is not left-orderable. Thus it is naturally conjectured that all non-\(L\)-space surgeries are left-orderable surgeries. In this paper, as an application of their construction, the first examples of hyperbolic knots are given such that all the non-trivial surgeries are non-\(L\)-space surgeries, which are left-orderable surgeries (Theorem 1.8). The key of the construction is using periodic knots. Actually, roughly speaking, it is shown that a periodic knot admits left-orderable surgeries if a factor knot does (Theorem 2.1), and that, under certain conditions, periodic knots have no \(L\)-space surgeries (Corollary 3.2).