On left-orderability and cyclic branched coverings

Tran, Anh T.

doi:10.2969/jmsj/06731169

Cited by 20 publications

(22 citation statements)

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“…(3) When ε = +1, we have 2(2k + 1)(2 + 1) + ε ≡ 3 (mod 4), and so [Hu15] implies that π 1 (Σ n (K [2(2k+1),2 +1] )) is left-orderable for all sufficiently large n. We do not know if the corresponding manifolds have co-orientable taut foliations or whether or not they are Lspaces. Arguments similar to those that appear in [Hu15] are used in [Tra15] to show (in particular) that π 1 (Σ n (K [2(2k+1),−(2 +1)] ) is also left-orderable for n sufficiently large. Moreover, in [Tra15] explicit lower bounds for n are given, both for ε = +1 and ε = −1.…”

Section: Some Excellent Cyclic Branched Covers Of Two-bridge Knotsmentioning

confidence: 99%

“…Arguments similar to those that appear in [Hu15] are used in [Tra15] to show (in particular) that π 1 (Σ n (K [2(2k+1),−(2 +1)] ) is also left-orderable for n sufficiently large. Moreover, in [Tra15] explicit lower bounds for n are given, both for ε = +1 and ε = −1. However, these bounds go to infinity as increases.…”

Section: Some Excellent Cyclic Branched Covers Of Two-bridge Knotsmentioning

confidence: 99%

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Taut Foliations, Left-Orderability, and Cyclic Branched Covers

Gordon

Lidman

2014

Acta Math Vietnam

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We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental groups, and are not L-spaces.2 CAMERON GORDON AND TYE LIDMAN mentioned above: if Y is an L-space, then Y does not admit a co-orientable taut foliation [OS04, KR15, Bow16].One reason why the possible equivalence of (1.1), (1.2), and (1.3) is desirable is that they exhibit different types of behavior. For example, left-orderability is well-understood under non-zero degree maps, while Floer homology is well-understood under Dehn surgery. The equivalence of (1.1), (1.2), and (1.3) would allow one to use these properties in tandem. We mention that it would follow from the equivalence of (1.1) and (1.3) that a toroidal integer homology sphere can never be an L-space (see [CLW13, Section 4]).Definition 1.1. A closed, connected, orientable three-manifold Y is excellent if Y admits a coorientable taut foliation and π 1 (Y ) is left-orderable. Consequently, Y is prime and is not an L-space. On the other hand, Y is called a total L-space if all three conditions (1.1), (1.2), and (1.3) fail. In other words, a total L-space is an L-space whose fundamental group is not left-orderable.We remark that if b 1 (Y ) > 0 and Y is prime, then Y always admits a co-orientable taut foliation [Gab83] and has left-orderable fundamental group [BRW05, Corollary 3.4], and thus is excellent. Many examples of total L-spaces have come from branched covers of knots, including all two-fold branched covers of non-split alternating links [BGW13, OS05b] and some higher-order branched covers of two-bridge knots [DPT05, Pet09]. In this paper, we study when cyclic branched covers of certain other families of knots give excellent manifolds or total L-spaces. We will denote the n-fold cyclic branched cover of a knot K by Σ n (K) and will always assume n ≥ 2. Note that by the Smith conjecture [MB84], Σ n (K) is simply-connected only if K is trivial.A key input throughout the paper is that, as mentioned above, the equivalence of (1.1), (1.2), and (1.3) is known for Seifert fibered manifolds. This enables us to give a complete analysis of the case of torus knots. Throughout, we let T p,q denote the (p, q)-torus link. Theorem 1.2. Let n, p, q ≥ 2 and suppose p, q are relatively prime. Then, Σ n (T p,q ) is excellent if and only if its fundamental group is infinite. Thus, Σ n (T p,q ) is excellent except in the cases: (i) {p, q} = {2, 3}, 2 ≤ n ≤ 5, (ii) {p, q} = {2, 5}, 2 ≤ n ≤ 3, (iii) {p, q} = {2, r}, r ≥ 7, n = 2, (iv) {p, q} = {3, 4}, n = 2, (v) {p, q} = {3, 5}, n = 2, in which case Σ n (T p,q ) is a total L-space.Both left-orderability and taut foliations have been studied in the context of gluing manifolds with toral boundary (see for instance [BC17, CLW13]). For this reason, we will also study certain families of satellite knots. While bordered Floer homology is suitable machinery for studying the Heegaard Floer homology of manifolds glued along surfaces [LOT08], we will not focus on its use here. Let K be a satellite knot with non-trivi...

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Section: Some Excellent Cyclic Branched Covers Of Two-bridge Knotsmentioning

confidence: 99%

Section: Some Excellent Cyclic Branched Covers Of Two-bridge Knotsmentioning

confidence: 99%

Taut Foliations, Left-Orderability, and Cyclic Branched Covers

Gordon

Lidman

2014

Acta Math Vietnam

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“…We prove Theorems 1.2 and 1.4 by studying representations of 3-manifold groups into the nonlinear Lie group G = PSL 2 R. Using such representations to order 3-manifold groups goes back at least to [EHN], and has been exploited repeatedly of late to provide evidence for Conjecture 1.1. Closest to our results here, representations to G were used to obtain ordering results for Dehn surgeries on two-bridge knots in [HT,Tra2], as well as branched covers of two-bridge knots in [Hu,Tra1]. Indeed, some of the results on branched covers in [Hu,Tra1,Gor] can be viewed as special cases of both the statement and the proof of Theorem 1.4(a).…”

Section: Translation Extension Locimentioning

confidence: 89%

Orderability and Dehn filling

Culler

Dunfield

2018

Geom. Topol.

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Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3-spheres. Specifically, for a compact 3-manifold M with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of M have left-orderable fundamental groups. Our technique uses certain representations from π 1 (M ) into PSL 2 R, which we organize into an infinite graph in H 1 (∂M ; R) called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.

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“…The proof of Step 1 is similar to that of [Tr,Proposition 2.7]. Suppose y ≤ 2 and Φ J(2m,2n) (x 0 , y) = 0.…”

Section: Proof Of Theorem 1(i)mentioning

confidence: 92%