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

Abstract

We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot K with property (D) with slope p q ≥ 2g(K) − 1 is not left orderable. By making full use of the fixed point method, we prove that (1) nontrivial knots which are closures of positive 1-bridge braids have property (D); (2) L-space satellite knots, with positive 1-bridge braid patterns, and companion with property (D), have property (D); (3) the fundamental group of the manifold obtained by Dehn filling on v2503 is not left orderable. Additionally, we prove that L-space twisted torus knots of form T l,m p,kp±1 are closures of positive 1-bridge braids.

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

“…And as its name suggests, any lens space is an L-space. It is also proved that any 1-bridge braid in the three-sphere or lens space is an L-space knot [9], and the L-spaces obtained by surgeries along 1-bridge braids in S 3 has non-left-orderable fundamental groups [17]. In line with these researches, we deduce similar properties of (1, 1) L-space knots in S 3 .…”

“…In this paper, we prove the following result in a similar way. Thanks to the additional symmetry, our proof is simplified compared to the proof of [17,Theorem 1.3]. Therefore, by [17,Theorem 4.1], we have the following conclusion.…”

“…The author proved that [17,Theorem 1.3] nontrivial knots which are closures of positive 1-bridge braids have property (D). And by [17,Theorem 4.1], it implies the non-leftorderability of the fundamental groups of the L-spaces obtained by Dehn surgeries on closures of 1-bridge braids. In this paper, we prove the following result in a similar way.…”

“…Because a (1, 1)-decomposition eases the computation of knot Floer homology, many examples of L-space knots which were studied in the literature are (1, 1)-knots. Theorem 5 serves as the generalization of relevant non-left-orderability results [3,4,11,12,13,14,15,16,17,22,23].…”

Slope detection and toroidal 3-manifolds

“…If o is a left-order on π 1 (U 0 ) with dynamic realisation ρ o : π 1 (U 0 ) → Homeo + (R), [Nie,Lemma 6.1] shows that ρ o (π 1 (∂U 0 )) has no fixed points in R. It follows from Lemma 6.5 that o is boundary-cofinal and that we can assume that ρ o (π 1 (∂U 0 )) ≤ Homeo Z (R). Lemma 6.7 then shows that o order-detects a slope.…”