2012
DOI: 10.1142/s0218216511010085
|View full text |Cite
|
Sign up to set email alerts
|

Instanton Floer Homology for Two-Component Links

Abstract: For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer-Mrowka instanton Floer homology for links.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1
1

Citation Types

0
12
0

Year Published

2013
2013
2017
2017

Publication Types

Select...
5

Relationship

2
3

Authors

Journals

citations
Cited by 7 publications
(12 citation statements)
references
References 14 publications
(31 reference statements)
0
12
0
Order By: Relevance
“…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%
“…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%
“…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%
“…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%