Strong L-spaces and left-orderability

Levine, Adam Simon; Lewallen, Sam

doi:10.4310/mrl.2012.v19.n6.a5

Cited by 10 publications

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“…, 2a k ] where a i > 0, for 1 ≤ i ≤ k, then Σ n (K) is the branched double cover of an alternating link [MV01]. Therefore, for any n ≥ 2, Σ n (K) is an L-space by [OS05b, Proposition 3.3] and π 1 (Σ n (K)) is not left-orderable [BGW13, Theorem 4] (see also [Gre11,Ito13,LL12]). The results of [Hu15,Tra15] say that for certain two-bridge knots K, we have that π 1 (Σ n (K)) is left-orderable for all sufficiently large n. The situation for cyclic branched covers of torus knots is described in Theorem 1.2.…”

Section: Total L-spaces Arising From Two-bridge Knotsmentioning

confidence: 99%

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: Total L-spaces Arising From Two-bridge Knotsmentioning

confidence: 99%

Taut Foliations, Left-Orderability, and Cyclic Branched Covers

Gordon

Lidman

2014

Acta Math Vietnam

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“…The sign pattern of the entries of a Pólya matrix is highly constrained. This fact plays a key role in the proof that the fundamental group of a strong L-space is not left-orderable [LL12]. Pólya matrices obey a deep structure theorem due independently to McCuaig [McC04] and Robertson, Seymour, and Thomas [RST99].…”

Section: Preliminariesmentioning

confidence: 97%

“…Both det(Y ) and rank( HF(Y )) provide lower bounds on M (Y ) by (1). The second author and Lewallen [LL12] introduced the following definition:…”

Section: Introductionmentioning

confidence: 99%

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Strong Heegaard diagrams and strong L–spaces

Greene

Levine

2016

Algebr. Geom. Topol.

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Abstract. We study a class of 3-manifolds called strong L-spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L-space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L-space admitting a strong Heegaard diagram of genus two via an explicit classification. We also show that there exist finitely many strong L-spaces with bounded order of first homology; for instance, through order eight, they are connected sums of lens spaces. The methods are topological and graph theoretic. We discuss many related results and questions.

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“…(v) Lewallen and Levine have shown that strong L-spaces do not have left-orderable fundamental groups [35].…”

Section: Introductionmentioning

confidence: 99%

Graph manifold ℤ-homology 3-spheres and taut foliations

Boileau

Boyer

2015

Journal of Topology

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We show that a graph manifold which is a double-struckZ‐homology 3‐sphere not homeomorphic to either S3 or Σ(2,3,5) admits a horizontal foliation. This combines with known results to show that the conditions of not being an L‐space, of having a left‐orderable fundamental group and of admitting a co‐oriented taut foliation, are equivalent for graph manifold double-struckZ‐homology 3‐spheres.

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Strong L-spaces and left-orderability

Cited by 10 publications

References 0 publications

Taut Foliations, Left-Orderability, and Cyclic Branched Covers

Taut Foliations, Left-Orderability, and Cyclic Branched Covers

Strong Heegaard diagrams and strong L–spaces

Graph manifold ℤ-homology 3-spheres and taut foliations

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