Constructing taut foliations

Roberts, Rachel

doi:10.1007/bf02566022

Cited by 31 publications

(28 citation statements)

References 20 publications

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“…We point out that the Li-Roberts Conjecture is known for many families of knots, including hyperbolic fibered knots [Rob01] and non-special alternating knots [Rob95].…”

Section: Introductionmentioning

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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“…We point out that the Li-Roberts Conjecture is known for many families of knots, including hyperbolic fibered knots [Rob01] and non-special alternating knots [Rob95].…”

Section: Introductionmentioning

confidence: 99%

Taut Foliations, Left-Orderability, and Cyclic Branched Covers

Gordon

Lidman

2014

Acta Math Vietnam

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“…Then Roberts' theorem [Ro,Theorem 2.3] can be stated as Theorem 5.1 (Roberts). IfB= (R^S) constructed above is an essential branched surface in E{K), and has no planar surface of contact, then B extends to an essential branched surface B 1 in if (7) for all slope 7 G J.…”

Section: Remark the Pictures Onmentioning

confidence: 99%

The classification of exceptional Dehn surgeries on 2-bridge knots

Brittenham¹,

Wu²

2001

Communications in Analysis and Geometry

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“…The arrows in the figure indicate the cusp direction of branched locus. A similar argument as in [14] shows that the branched surface fully carries a lamination with no compact leaves and each negative slope can be realized as the boundary slope of a lamination fully carried by B. Since the disk D intersects the knot exactly twice, the branched surface B is essential in M by [9, 3.12].…”

Section: Proof Of Propositionmentioning

confidence: 94%

“…We first construct an essential branched surface B in the exterior M and then prove that B (which has boundary on ∂M ) can be capped off by a branched surface in V to yield an essential branched surfaceB in M (−1). The construction of B is similar to that given in [14].…”

Section: Proof Of Propositionmentioning

confidence: 99%