$L$-space surgeries on links

Liu, Yajing

doi:10.4171/qt/96

Cited by 18 publications

(62 citation statements)

References 51 publications

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“…In terms of the large surgery, Gorsky and Némethi defined L-space links in [3]. [7]. By Equation (2.1) and the inclusionexclusion formula, one can write [1]:…”

Section: 3mentioning

confidence: 99%

“…Remark 2.18. For L-space links with two components, the explicit formula for the Hfunction can also be found in [7].…”

Section: 3mentioning

confidence: 99%

“…Thus for sufficiently large g i , g ∈ G(L). The H-function of L k is computed in [7,Proposition 6.12]. Then the h-function of L k is shown in Figure 4, where h(k, 0) = 0, and h(k − 1, 0) = 1.…”

Section: 3mentioning

confidence: 99%

“…Therefore, G(L k ) = G HF (L k ). [7]. Its Alexander polynomial equals ∆ L (t 1 , t 2 , t 3 ) = (t 1/2 Each component L i is an unknot and bounds a disk D i in B 4 .…”

Section: 3mentioning

confidence: 99%

“…Let k be a positive integer. The two bridge link L k = b(4k 2 + 4k, −2k − 1) is a 2-component L-space link with linking number 0[7], and both link components are unknots, seeFigure 3. The Alexander polynomial of L k is computed in[7, Section 3],∆ L k (t 1 , t 2 ) = (−1) k |i+1/2|+|j+1/2|≤k (−1) i+j t…”

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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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“…In terms of the large surgery, Gorsky and Némethi defined L-space links in [3]. [7]. By Equation (2.1) and the inclusionexclusion formula, one can write [1]:…”

Section: 3mentioning

confidence: 99%

“…Remark 2.18. For L-space links with two components, the explicit formula for the Hfunction can also be found in [7].…”

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

“…Therefore, G(L k ) = G HF (L k ). [7]. Its Alexander polynomial equals ∆ L (t 1 , t 2 , t 3 ) = (t 1/2 Each component L i is an unknot and bounds a disk D i in B 4 .…”

Section: 3mentioning

confidence: 99%

mentioning

confidence: 99%

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Four-genera of links and Heegaard Floer homology

Liu

2019

Algebr. Geom. Topol.

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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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Introduction to Heegaard Floer homology

Gorsky¹

2019

Russ. Math. Surv.

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$L$-space surgeries on links

Cited by 18 publications

References 51 publications

Four-genera of links and Heegaard Floer homology

Four-genera of links and Heegaard Floer homology

On the set of L-space surgeries for links

Introduction to Heegaard Floer homology

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