Left-orderable fundamental group and Dehn surgery on the knot $5_2$

Abstract

We show that the resulting manifold by r-surgery on the knot 5 2 , which is the two-bridge knot corresponding to the rational number 3/7, has left-orderable fundamental group if the slope r satisfies 0 ≤ r ≤ 4.

Left-orderable fundamental groups and Dehn surgery on genus one 2–bridge knots

“…Later, Clay, Lidman and Watson [6] showed r = ±4 are also leftorderable. These results were extended to all hyperbolic twist knots [10,11,19]. Also, Tran [20] further extended the range of left-orderable slopes for twist knots.…”

On left-orderable fundamental groups and Dehn surgeries on knots

“…Recently, Hakamata and Teragaito have generalized this result to all hyperbolic twist knots. They show that if 0 ≤ r ≤ 4 then r-surgery on any hyperbolic twist knot yields a manifold whose fundamental group is left-orderable [HT1,HT2]. In this paper, we study the left-orderability of the fundamental group of manifolds obtained by Dehn surgeries on a large class of two-bridge knots that includes all twist knots.…”

On left-orderability and cyclic branched coverings

“…This conjecture predicts that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. The conjecture has been confirmed for Seifert fibered manifolds, Sol manifolds, double branched coverings of non-splitting alternating links [BGW] and Dehn surgeries on the figure eight knot, on the knot 5 2 and more generally on genus one two-bridge knots (see [BGW, CLW], [HT1] and [HT2,HT3,Tr] respectively). A technique that has so far worked very well for proving the left-orderability of fundamental groups is lifting a non-abelian SU(1, 1) representation (or equivalently a non-abelian SL 2 (R) representation) of a 3-manifold group to the universal covering group SU(1, 1) and then using the result by Bergman [Be] that SU(1, 1) is a left-orderable group.…”

“…A technique that has so far worked very well for proving the left-orderability of fundamental groups is lifting a non-abelian SU(1, 1) representation (or equivalently a non-abelian SL 2 (R) representation) of a 3-manifold group to the universal covering group SU(1, 1) and then using the result by Bergman [Be] that SU(1, 1) is a left-orderable group. This technique, which is based on an important result of Khoi [Kh], was first introduced in [BGW] and was applied in [HT1,HT2,HT3,Tr] to study the left-orderability of Dehn surgeries on genus one two-bridge knots.…”