L-spaces, taut foliations, and graph manifolds

Hanselman, Jonathan; Rasmussen, Jacob; Rasmussen, Sarah Dean; Watson, Liam

doi:10.1112/s0010437x19007814

Cited by 29 publications

(31 citation statements)

References 17 publications

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“…For instance, it is known that the conjectures hold for graph manifolds ( and ) and since the 2‐fold branched covers of arborescent links are of this type, we deduce: Proposition Let

L

be an arborescent link and suppose that either

{normalΣ}_{2} false(L false)

is not CTF or

{normalΣ}_{2} false(L false)

is not LO. (1)If

L

is strongly quasi‐positive, then

g_{4}^{t o p} false(L false) = g false(L false) = \frac{1}{2} false(false| σ (L) false| - (m - 1) false)

. (2)If

L

is quasi‐positive, then

g_{4}^{t o p} false(L false) = g_{4} false(L false) = \frac{1}{2} false(false| σ (L) false| - (m - 1) false)

. …”

Section: Questions Problems and Remarksmentioning

confidence: 90%

Branched covers of quasi‐positive links and L‐spaces

Boileau

Boyer

Gordon

2019

Journal of Topology

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Let L be an oriented link such that Σn(L), the n-fold cyclic cover of S 3 branched over L, is an L-space for some n 2. We show that if either L is a strongly quasi-positive link other than one with Alexander polynomial a multiple of (t − 1) 2g(L)+(|L|−1) , or L is a quasi-positive link other than one with Alexander polynomial divisible by (t − 1) 2g 4 (L)+(|L|−1) , then there is an integer n(L), determined by the Alexander polynomial of L in the first case and the Alexander polynomial of L and the smooth 4-genus of L, g4(L), in the second, such that n n(L). If K is a strongly quasi-positive knot with monic Alexander polynomial such as an L-space knot, we show that Σn(K) is not an L-space for n 6, and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if Σn(K) is an L-space for some n = 2, 3, 4, 5. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating quasi-positive links. They also allow us to classify strongly quasi-positive alternating links and 3-strand pretzel links.

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“…For instance, it is known that the conjectures hold for graph manifolds ( and ) and since the 2‐fold branched covers of arborescent links are of this type, we deduce: Proposition Let

L

be an arborescent link and suppose that either

{normalΣ}_{2} false(L false)

is not CTF or

{normalΣ}_{2} false(L false)

is not LO. (1)If

L

is strongly quasi‐positive, then

g_{4}^{t o p} false(L false) = g false(L false) = \frac{1}{2} false(false| σ (L) false| - (m - 1) false)

. (2)If

L

is quasi‐positive, then

g_{4}^{t o p} false(L false) = g_{4} false(L false) = \frac{1}{2} false(false| σ (L) false| - (m - 1) false)

. …”

Section: Questions Problems and Remarksmentioning

confidence: 90%

Branched covers of quasi‐positive links and L‐spaces

Boileau

Boyer

Gordon

2019

Journal of Topology

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“…This implies that the answer to Question 1.7 is "yes," provided that Y has Seifert geometry. If one works with Z/2Z-coefficients, the same holds for Sol geometry [BRW05,BGW13] and graph manifolds [BC17,HRRW15].…”

Section: Introductionmentioning

confidence: 89%

Floer homology and covering spaces

Lidman

Manolescu²

2018

Geom. Topol.

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We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p n -sheeted regular cover that is a Z/pZ-L-space (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.

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“…‡ The author conceived this foliation-classification project [29] shortly before her summons to collaborate with Hanselman et al [14]. These two proofs of the graph-manifold L-space conjecture make contact with foliations via disparate mechanisms.…”

Section: Torus-link Satellitesmentioning

confidence: 99%

“…Despite reliance on an enhanced L-space gluing tool proved in Theorem 3.6, † this paper was primarily made possible by the author's classification of graph manifolds admitting co-oriented taut foliations, with proof of the graph-manifold Lspace conjecture as by-product [29]. (This is not to be confused with the author's joint work with Hanselman et al [14].) ‡ This classification combines a new classification formula (Theorem 4.3), generalizing that of Jankins and Neumann for Seifert fibered spaces [19], with a structure theorem (Theorem 4.4) prescribing the interpretation of outputs of this formula.…”

Section: Introductionmentioning

confidence: 99%

Rational L‐space surgeries on satellites by algebraic links

Rasmussen

2020

Journal of Topology

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Given an n-component link L in any 3-manifold M , the space L ⊂ (Q ∪ {∞}) n of rational surgery slopes yielding L-spaces is already fully characterized in joint work by the author when n = 1 and L is nontrivial. For n >1, however, there are no previous results for L as a rational subspace, and only limited results for integer surgeries L ∩ Z n on S 3. Herein, we provide the first nontrivial explicit descriptions of L for rational surgeries on multi-component links. Generalizing Hedden's and Hom's L-space result for cables, we compute both L, and its topology, for all satellites by torus-links in S 3. For fractal-boundaried L resulting from satellites by algebraic links or iterated torus links, we develop arbitrarily precise approximation tools. We also extend the provisional validity of the L-space conjecture for rational surgeries on a knot K ⊂ S 3 to rational surgeries on such satellite-links of K. These results exploit the author's generalized Jankins-Neumann formula for graph manifolds.

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L-spaces, taut foliations, and graph manifolds

Abstract: If Y is a closed orientable graph manifold, we show that Y admits a coorientable taut foliation if and only if Y is not an L-space. Combined with previous work of Boyer and Clay, this implies that Y is an L-space if and only if π1(Y ) is not left-orderable.

Cited by 29 publications

References 17 publications

Branched covers of quasi‐positive links and L‐spaces

Branched covers of quasi‐positive links and L‐spaces

Floer homology and covering spaces

Rational L‐space surgeries on satellites by algebraic links

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