Instanton Floer Homology for Two-Component Links

Harper, Eric; Saveliev, Nikolai

doi:10.1142/s0218216511010085

Cited by 7 publications

(12 citation statements)

References 14 publications

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“…Thus, if the linking number is non-zero then Theorem 1.2 asserts the existence of a flat connection in the non-trivial SO(3) bundle over Y . This instance of the theorem can also be deduced from a result of Harper-Saveliev [10]. It clearly suffices to prove Theorems 1.1 and 1.2 for r = b 1 (X).…”

Section: Introductionmentioning

confidence: 69%

On the existence of representations of finitely presented groups in compact Lie groups

Froyshov

2013

Topology and its Applications

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Section: Introductionmentioning

confidence: 69%

On the existence of representations of finitely presented groups in compact Lie groups

Froyshov

2013

Topology and its Applications

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“…The result then follows from [21,Theorem 2.3] which asserts that the Euler characteristic of the sutured Floer homology of L equals ± ℓk (ℓ 1 , ℓ 2 ). Theorem 8.1 implies in particular that the Euler characteristic of I ♮ (k) equals ±1, which is the linking number of the two components of the link k ♮ .…”

Section: Knot Homology: Explicit Calculationsmentioning

confidence: 92%

“…The desired formula now follows because, according to the Gauss-Bonnet theorem, in the formula ind D τ θ (X) = −1, which corresponds to the fact that the (+1)-eigenspace of the involutionτ * : H 0 (X; ad θ) → H 0 (X; ad θ) is onedimensional. 21 3.7. Proof of Lemma 2.5.…”

Section: Equivariant Indexmentioning

confidence: 99%

Link homology and equivariant gauge theory

Poudel

Saveliev

2017

Algebr. Geom. Topol.

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The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, torus knots, and Montesinos knots, as well as for several families of two-components links. IntroductionThis paper studies the Floer homology I * (Σ, L) of two-component links L ⊂ Σ in homology 3-spheres defined by Kronheimer and Mrowka [26] using singular SO (3) instantons. An important special case of this theory is the singular instanton knot Floer homology I ♮ (k) for knots k ⊂ S 3 obtained by applying I * (S 3 , L) to the link L which is a connected sum of k with the Hopf link. The Floer homology I * (Σ, L) has a relative Z/4 grading, which can be upgraded to an absolute Z/4 grading in the special case of I ♮ (k). [26] used I ♮ (k) and its close cousin I ♯ (k) to prove that the reduced Khovanov homology is an unknot-detector. Kronheimer and MrowkaThe definition of groups I * (Σ, L) uses singular gauge theory, which makes them difficult to compute. We propose a new approach to these computations which uses equivariant gauge theory in place of the singular one.Given a two-component link L in an integral homology sphere Σ, we pass 2010 Mathematics Subject Classification. 57M27, 57R58.Both authors were partially supported by NSF Grant 1065905. 1 to the double branched cover M → Σ with branch set L and observe that the singular connections on Σ used in the definition of I * (Σ, L) pull back to equivariant smooth connections on M . The generators of the Floer chain complex IC * (Σ, L), whose homology is I * (Σ, L), are then derived from the equivariant representations π 1 M → SO(3), and their Floer gradings are computed using equivariant rather than singular index theory † .As our first application of this approach, we determine the grading of the special generator in the Floer chain complex IC ♮ (k) of a knot k ⊂ S 3 , see Section 5. This fixes the absolute Z/4 grading on I ♮ (k) and confirms the conjecture of Hedden, Herald and Kirk [20].Theorem. For any knot k ⊂ S 3 , the grading of the special generator in the Floer chain complex IC ♮ (k) equals sign k mod 4.We also achieve significant simplifications in computing the Floer chain complexes IC ♮ (k) and IC * (Σ, L) for knots and links with simple double branched covers, such as torus and Montesinos knots and links, whose double branched covers are Seifert fibered manifolds. Explicit calculations for these knots and links are possible because the gauge theory on Seife...

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“…We choose ε ∈ (Z N ) k compatible with the given labels, which means that ε satisfies Equation (8) with respect to the braid σ , namely i∈I ξ j ε i = ω a j holds for j = 1, . .…”

Section: 4mentioning

confidence: 99%

“…The second author and N. Saveliev used projective SU (2) representations to extend the Casson-Lin invariant to 2-component links L in S 3 in [7], and they showed that h(L) = ±lk( 1 , 2 ), the linking number of L = 1 ∪ 2 . They gave a gauge theoretic description of the invariant h(L) in [8], where they also described Floer homology groups with Euler characteristic equal to h (L).…”

Section: Introductionmentioning

confidence: 99%

The SU(N) Casson–Lin invariants for links

Bodén

Harper²

2016

Pacific J. Math.

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We introduce the SU (N ) Casson-Lin invariants for links L in S 3 with more than one component. Writing L = 1 ∪ · · · ∪ n , we require as input an n-tuple (a 1 , . . . , a n ) ∈ Z n of labels, where a j is associated with j . The SU (N ) Casson-Lin invariant, denoted h N,a (L), gives an algebraic count of certain projective SU (N ) representations of the link group π 1 (S 3 L), and the family h N,a of link invariants gives a natural extension of the SU (2) Casson-Lin invariant, which was defined for knots by X.-S. Lin and for 2-component links by Harper and Saveliev. We compute the invariants for the Hopf link and more generally for chain links, and we show that, under mild conditions on the labels (a 1 , . . . , a n ), the invariants h N,a (L) vanish whenever L is a split link.

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Instanton Floer Homology for Two-Component Links

Cited by 7 publications

References 14 publications

On the existence of representations of finitely presented groups in compact Lie groups

On the existence of representations of finitely presented groups in compact Lie groups

Link homology and equivariant gauge theory

The SU(N) Casson–Lin invariants for links

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