An explicit description of $(1,1)$ L-space knots, and non-left-orderable surgeries

Abstract

Greene, Lewallen and Vafaee characterized (1, 1) L-space knots in S 3 and lens space in the notation of coherent reduced (1, 1)-diagrams. We analyze these diagrams, and deduce an explicit description of these knots. With the new description, we prove that any L-space obtained by Dehn surgery on a (1, 1)-knot in S 3 has non-left-orderable fundamental group.

Order-detection of slopes on the boundaries of knot manifolds

“…In this section we interpret boundary-cofinality in terms of actions on the reals to connect it with work of Nie ([Nie1,Nie2]). (1) γ ∈ π 1 (M ) is o-cofinal if and only if ρ o (γ) acts fixed-point freely on R.…”

“…As examples, L-space knot exteriors are known to be Floer simple, and among such knots it is known that non-trivial torus knot exteriors have the property that D wk ord (M ) = S(M ) and that the same is true for certain pretzel knots, twisted torus knots and cables of such knots ( [BC,CW1,CW2]). From Corollary 6.3, we know that the exteriors of the closures of a 1-bridge braids ([Nie1]), which are L-space knots, and (1, 1) L-space knot exteriors ([Nie2]) provide further examples.…”