Left-orderablity for surgeries on (−2,3,2s + 1)-pretzel knots

Abstract

In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a (−2, 3, 2s + 1)-pretzel knot (s ≥ 3) with slope p q is not left orderable if p q ≥ 2s + 3, and that it is left orderable if p q is in a neighborhood of zero depending on s.

On $1$-bridge braids, satellite knots, the manifold $v2503$ and non-left-orderable surgeries and fillings

“…A number of research papers [5,6,22,23,25,28,30,32] have been devoted to study non-leftorderable surgeries on subfamilies of the L-space twisted torus knots T l,m p,kp±1 . And recently, Tran proved [33] that the fundamental group of the manifold obtained by Dehn surgery on an L-space twisted torus knot of form T l,m p,kp±1 with slope at least 2g(T l,m p,kp±1 ) − 1 is not left orderable.…”

Taut Foliations, Positive 3-Braids, and the L-Space Conjecture

“…Tran, generalizing work of Nie [Nie18], showed that for any K in an infinite subfamily F of 3-stranded twisted torus knots, and r ≥ 2g(K) − 1, π 1 (S 3 r (K)) is not left-orderable [Tra18]. The L-space pretzel knots comprise a proper subset of F. We conclude:…”

Promoting circular-orderability to left-orderability

“…Example 3.4. -Let K be a fibred hyperbolic knot in S 3 for which p qsurgery yields a manifold K( p q ) with non-left-orderable fundamental group whenever p q r for some r ∈ R. Examples of such knots are provided by [10,14] and others, but for the sake of concreteness we may take K to be the (−2, 3, 2n + 1)-pretzel knot where n 3, in which case r = 2n + 3 by [32].…”