Cable links and L-space surgeries

Gorsky, Eugene; Hom, Jennifer

doi:10.4171/qt/98

Cited by 12 publications

(15 citation statements)

References 9 publications

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“…The Heegaard Floer link homology. Ozsváth and Szabó associated the multigraded link invariants HF L − (L) and HF L(L) to links L ⊂ S 3 where HF L − (L) was defined in Equation (2.1), and HF L(L) is defined as follows [2,16]:…”

Section: 3mentioning

confidence: 99%

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Four-genera of links and Heegaard Floer homology

Liu

2019

Algebr. Geom. Topol.

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For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint surfaces in B 4 . In this paper, we describe the set of genera of such surfaces in terms of the h-function, which is a link invariant from Heegaard Floer homology. In particular, we use the h-function to give lower bounds for the 4-genus of the link. For L-space links, the h-function is explicitly determined by Alexander polynomials of the link and sublinks. We show some L-space links where the lower bounds are sharp, and also describe all possible genera of disjoint surfaces bounded by such links.

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Section: 3mentioning

confidence: 99%

“…If L is an L-space link, there exist spectral sequences converging to HF L − (L) and HF L(L) respectively [2,3].…”

Section: 3mentioning

confidence: 99%

Four-genera of links and Heegaard Floer homology

Liu

2019

Algebr. Geom. Topol.

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“…Since the right-hand condition of (265) is equivalent to the condition that Y Γ (y Γ ) be an L-space, the Claim proves that Y Γ (y Γ ) is an L-space if and only if Y Γ (y Γ ) = S 3 . Thus, if we define Λ Γ as in (11), then the statement L(Y Γ ) = Λ Γ holds tautologically.…”

Section: L-space Surgery Regions For Iterated Satellites: Proof Of Thmentioning

confidence: 99%

“…This phenomenon also affects Λ Γ as defined in (11). While Λ Γ ⊃ v∈Vert(Γ) Λ v , this containment is proper if there is an edge e ∈ Edge(Γ) for which one can have Y Γ v(−e) (y Γ v(−e) ) bc with 0 = y v(e) j(e)± ∈ Z if j(e) = −1, or any integer value y v(e) j(e)± ∈ Z if j(e) = −1.…”

Section: Exceptional Symmetriesmentioning

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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“…Suppose that K is an L-space knot, m and n are positive coprime integers and n/m > 2g(K) − 1. By [9], the two-component cable link K 2m,2n is an L-space link. Then the set LS(K 2m,2n ) is unbounded from below.…”

Section: Introductionmentioning

confidence: 99%

On the set of L-space surgeries for links

Gorsky

Némethi

2018

Advances in Mathematics

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It it known that the set of L-space surgeries on a nontrivial L-space knot is always bounded from below. However, already for two-component torus links the set of L-space surgeries might be unbounded from below. For algebraic two-component links we provide three complete characterizations for the boundedness from below: one in terms of the h-function, one in terms of the Alexander polynomial, and one in terms of the embedded resolution graph. They show that the set of L-space surgeries is bounded from below for most algebraic links. In fact, the used property of the h-function is a sufficient condition for non-algebraic L-space links as well.be a link with r components. We define LS(L) ⊂ Z r to be the set of all r-tuples (d 1 , . . . , d r ) such that the surgery S 3 d 1 ,...,dr (L) of S 3 along L with coefficients (d 1 , . . . , d r ) is an L-space.By definition, L is an L-space link if and only if (Z ≥N ) r ⊂ LS(L) for some N. The structure of the set LS for knots is described by the following result. Theorem 1.1.2. ([31, 33], [14, Lemma 2.13]) Let K be a nontrivial L-space knot. Then S 3 d (K) is an L-space if and only if d ≥ 2g(K) − 1. In other words, LS(K) = [2g(K) − 1, +∞).On the other hand, already for two-component links the structure of the set LS becomes very complicated. For example, the sets LS(T (2p, 2q)) for two-component torus links were studied for p = 1 in [21] and for p > 1 in [10], and happen to be unbounded from below (see Figure 1 for the structure of LS for the (4, 6) torus link). In this paper, we study the following basic question about L-space links.Problem 1.1.3. For which L-space links the set LS(L) is bounded from below?Note that by a theorem of Liu [21] the Heegaard-Floer homology of any surgery on a 2component L-space link is completely determined by its Heegaard-Floer link homology, which, in its turn, is determined by the bivariate Alexander polynomial. However, it appears to be hard to use this algorithm directly to determine the set LS(L). We give the following partial answer.Assume that L has 2 components. Let h be the h-function for L (defined in [11]), h i are the h-functions for L i and v * is the point naturally dual to v, see Definition 3.4.

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Cable links and L-space surgeries

Cited by 12 publications

References 9 publications

Four-genera of links and Heegaard Floer homology

Four-genera of links and Heegaard Floer homology

Rational L‐space surgeries on satellites by algebraic links

On the set of L-space surgeries for links

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