Knot Floer homology, link Floer homology and link detection

We give new link detection results for knot and link Floer homology inspired by recent work on Khovanov homology. We show that knot Floer homology detects $T(2,4)$, $T(2,6)$, $T(3,3)$, $L7n1$, and the link $T(2,2n)$ with the orientation of one component reversed. We show link Floer homology detects $T(2,2n)$ and $T(n,n)$, for all $n$. Additionally we identify infinitely many pairs of links such that both links in the pair are each detected by link Floer homology but have the same Khovanov homology and knot Floer homology. Finally, we use some of our knot Floer detection results to give topological applications of annular Khovanov homology.


Introduction
Knot and link Floer homology are invariants of links in S 3 [24,27,25].There are a number of formal similarities between these Floer theoretic invariants and the combinatorial theory Khovanov homology.Recently, Khovanov homology has been shown to detect a number of simple links [30,20,32,17,4].Some of these detection results have used knot and link Floer homology without going so far as to determine whether knot or link Floer homology detects the relevant link.Inspired by this work, we give such detection results for knot and link Floer homology.We remind the reader that the knot Floer homology of a link L is computed using an associated knot, called the knotification of L, in a connected sum of S 1 × S 2 's while the link Floer homology of L is computed directly from the link L in S 3 .
Previously it was known that knot Floer homology detects the unknot [23], the trefoil [6], the figure eight knot [6], the Hopf link [21,23], and the unlink [22,12].Link Floer homology was known to detect the trivial 1 n-braid together with its braid axis [2] and determine if a link is split [29].It was also known that a stronger version of link Floer homology CFL ∞ detects the Borromean rings and the Whitehead link [8].
Throughout, we take the links T (2, 2n) to be oriented as the closure of the 2-braids σ 2n 1 .
Let J n be the link obtained from T (2, 2n) by reversing the orientation on one of the components.We have the following result: Theorem 3.1.If HFK(L) ∼ = HFK(J n ) for some n, then L is isotopic to J n .
We also prove the following Link Floer homology detection results: Theorem 3.2.If HFL(L) ∼ = HFL(T (2, 2n)) for some n, then L is isotopic to T (2, 2n).A consequence of these detection results is that every link currently known to be detected by Khovanov homology is also detected by either knot or link Floer homology.This leads to the following natural question; Question 1.1.Is there a link which Khovanov homology detects but which neither knot nor link Floer homology detects?
On the other hand, we show that there are infinitely many links detected by link Floer homology but which are detected by neither Khovanov homology nor knot Floer homology.
Finally, we use some of our torus link detection results to derive applications to annular Khovanov homology.Annular Khovanov homology is an invariant of links in the thickened annulus A × I, sometimes thought of as S 3 \ U where U is an unknot or the annular axis.To do this we utilize a generalization of the Ozsváth-Szabó spectral sequence, which relates annular Khovanov homology and knot Floer homology of the lift of the annular axis Ũ in Σ(L), the double branched cover of L [28,11].
This paper is organised as follows; in Section 2 we briefly review knot and link Floer homology.In section 3 we prove that knot Floer homology detects J n and link Floer homology detects T (2, 2n).In section 4 we prove that knot Floer homology detects T (2, 4).In section 5 we prove that knot Floer homology detects T (2,6).In section 6 we prove that link Floer homology detects T (n, n).In section 7 we prove that knot Floer homology detects T (3, 3).In section 8 we prove that knot Floer homology detects L7n1.In section 9 we prove that there are infinite families of links dectected by link Floer homology that also have the same Khovanov homology and knot Floer homology.Finally, in section 10 we prove the annular Khovanov homology results using some of our knot Floer detection results.
paper and suggesting the arguments involving link symmetries in Section 10, and Daniel Ruberman for helpful conversations about symmetries of Seifert fibered links.

Knot Floer homology and link Floer homology
Knot Floer homology and link Floer homology are invariants of links in S 3 defined using a version of Lagrangian Floer homology [24,27,25].They are categorifications of the single variable and multivariable Alexander polynomials respectively.Here we briefly highlight the key features of knot Floer homology and link Floer homology that we use to obtain our detection results.Throughout this paper we work with coefficients in Z/2Z.
Let L be an oriented link in S 3 , with components L 1 , L 2 , . . ., L n .The link Floer homology of L is a multi-graded vector space The grading denoted by "d" above is called the Maslov or algebraic grading, while the A i gradings are called the Alexander gradings.Each The knot Floer homology of L is a vector space bi-graded by a Maslov grading and a single Alexander grading.The knot Floer homology of L can be obtained by projecting the link Floer homology complex onto the diagonal of the multi-Alexander gradings, which becomes the Alexander grading, and adding n−1 2 to the Maslov grading.
We use a number of formal properties of knot and link Floer homology in proving our link detection results.The first of these is that link Floer homology has a symmetry relating the component of the complex supported in grading (m, A 1 , . . ., A n ) with the component of the complex supported in grading Knot Floer homology enjoys the same symmetry property, since it can be defined by projecting the multi-Alexander gradings onto the diagonal.There is also a Künneth formula for computing the link Floer homology of a connected sum in terms of a tensor product of link Floer homologies.
The main formal property we will use, however, is that the link Floer homology of L admits spectral sequences to the link Floer homologies of its sublinks [2, Lemmas 2.2 and 2.3], [25].In particular, when L i is a sublink of L there is a spectral sequence from HFL(L) to HFL(L − L i ) ⊗ V |Li| shifting each Alexander grading by 1 2 ℓk(L j , L i ).It follows that there is also a spectral sequence from HFL(L) to HF(S 3 ) ⊗ V n−1 , or equivalently that there is a spectral sequence from HFK(L) to HF(# n−1 (S 1 × S 2 )).Here V is the multigraded vector space F ⊕ F with non-zero Maslov gradings 0 and −1 and multi-Alexander grading (0, . . ., 0).
In addition to enjoying the above algebraic properties, HFK(L) and HFL(L) are known to reflect a number of topological properties of L. For starters, there is a number of things we can say about the number of components of L. Since HFL(L) admits a spectral sequence to Moreover, since knot Floer homology categorifies the Alexander-Conway polynomial, and the Alexander-Conway polynomial polynomial number of two component links by a result of Hoste [13], it follows that knot Floer homology detects the linking number of two component links.
We will also make use of the fact that the link Floer homology of L yields information about the topology of S 3 − L; in particular that link Floer homology detects the Thurston norm of S 3 − L [26].Finally, if L is not a split link then the top Alexander grading associated to a component L i determines if L − L i is a braid in the complement of L i .Specifically, this happens exactly when the rank in maximal non-zero Alexander 3 Knot Floer homology detects J n Given that knot Floer homology detects the genus of a link, it is natural to try and detect links of small genus.The one component case is that of the unknot, which knot Floer homology is known to detect.The two component case is that of the 2-cable links.The simplest of these are 2-cables of unknots, i.e. the links T (2, 2n) with the orientation of one component reversed.We call these links J n .In this section we show that knot Floer homology detects each J n .
Theorem 3.1.If HFK(L) ∼ = HFK(J n ) for some n, then L is isotopic to J n .
For reference, we note that when n is positive HFK( , where the subscript denotes the Maslov grading of the generator, and [i] denotes the Alexander grading of a summand.When n is negative then HFK(J n ) can be computed from the above formula using an understanding of how knot Floer homology is affected by mirroring.
A consequence of this detection result is that link Floer homology detects the links T (2, 2n).This follows from Theorem 3.1 by considering how link Floer homology changes under reversing the orientation of a single component.
To prove Theorem 3.1, we will first prove that L is a 2-component link and that both of the components of L are unknots.Then we will use the fact that knot Floer homology detects genus to show that J n is detected among 2-component links with unknotted components.The topological argument used here was communicated to the authors by Eugene Gorsky To see that both components of L are unknotted, we consider the spectral sequences from HFK(L) to From this spectral sequence we see that HFK(K) is zero in all Maslov gradings except possibly 0 and 1. Considering how the Maslov grading changes under the symmetry of the Alexander grading for knot Floer homology we can see that HFK(K) can only be supported in Alexander grading 0, so K is an unknot.
With Lemma 3.3, we are now ready to prove Theorem 3.1.The key step is to deduce that J n is a cable of the unknot.
Proof of Theorem 3.1.Considering the Alexander grading of HFK(L), we see that the two components of L bound an annulus.Thus L is the twisted 2-cable of some knot.Each component of L is isotopic to the knot that was cabled and so L is a twisted 2-cable of the unknot.This means that L is J m for some m.Finally, a simple computation of the respective ranks in each Maslov grading shows that HFK(J m ) ∼ = HFK(J n ) if and 4 Knot Floer homology detects T (2, 4) Here we will utilize the results of the previous section to obtain a detection result for the torus link T (2, 4).
The link Floer homology of T (2, 4) is shown in Table 1, for reference.
To prove this we show the following lemma: has a unique Maslov index 0 generators.Moreover, that generator is supported in Alexander grading 0 in We then show that L has the same link Floer homology as T (2, 4) using structural properties of link Floer homology and apply Theorem 3.2 to complete the proof.
The following lemma will be useful in proving Lemma 4.2: at most zero with a unique Maslov grading zero generator.Then there is a unique Maslov index grading 0 generator in HFK(K), and it is of non-negative Alexander grading.
Proof.The Maslov grading 0 generator must persist under the spectral sequence from HFL(L) to HFL(K) ⊗ V |L|−1 , as else it cannot persist to HF(S 3 ) ⊗ V |L|−1 .If this generator sat in a negative Alexander grading 1 The link Floer homology of T (2, 4).The coordinates give the multi-Alexander grading, the subscript gives the Maslov grading.
then the symmetry properties of knot Floer homology would imply that there is a positive Maslov index generator in HFL(K) ⊗ V |L|−2 .However there are no positive Maslov index generators in HFL(L) and so there are none in HFL(K) With this in hand we can prove Lemma 4.2: Proof of Lemma 4.2.Suppose L is an n component link such that HFK(L) ∼ = HFK(T (2, 4)).Then n ≤ 2 since HFK(L) admits a spectral sequence to HF(# n−1 S 1 × S 2 ).Indeed, since rank( HFK(L)) is odd for knots, we have that n = 2. Since knot Floer homology detects the linking number of two component links, There is only one generator in Maslov grading 0 and it must survive in the spectral sequences from HFL(L) to HFL(L i ) ⊗ V .We call this generator θ 0 .The bi-Alexander grading for θ 0 is then ( where A i is the Alexander grading of the generator in Maslov grading 0 in HFK(L i ) and l is the linking number between the components.Since To complete our proof of Theorem 4.1 we show that if L has the same knot Floer homology as T (2, 4) then L also has same link Floer homology as T (2,4).This result combined with Theorem 3.2 proves Theorem 4.1.
Proof of Theorem 4.1.Suppose L is a link such that HFK(L) ∼ = HFK(T (2, 4)).We seek to understand HFL(L).From the argument in the proof of Lemma 4.2 we know that the only Maslov grading 0 generator of HFL(L) sits in bi-Alexander grading (1, 1).
Since there are spectral sequences from HFL(L) to HFK(K i ) ⊗ V for each i, we see that there are also generators of HFL(L) in (A 1 , A 2 ) gradings (1, 0) and (0, 1).The symmetry of HFK(L) gives generators at (−1, −1), (−1, 0) and (0, −1) as well.The are now two more generators to add so that the link Floer homology has rank 8. To maintain an even rank in each A i grading, they both must be added at the same bi-grading.
The only way to do this and maintain symmetry is to add them at (0, 0) so that HFL(L) ∼ = HFL(T (2, 4)) and Theorem 3.2 shows L is isotopic to T (2, 4).
5 Knot Floer homology detects T (2, 6) In the previous section we showed that knot Floer homology detects the torus link T (2, 4).The torus link ) is then a natural candidate for detection results.In this section we show that knot Floer homology indeed detects T (2, 6).
Since HFK(L) admits a spectral sequence to HF(# n−1 S 1 × S 2 ), where n is the number of components of L, L has at most two components.Indeed, as rank( HFK(L)) is even, L has exactly two components.Since knot Floer homology detects the linking number of two component links, the linking number is three.
From here the proof of Theorem 5.1 amounts to an algebraic argument showing that HFL(L) ∼ = HFK(L), and applying Theorem 3.2.
For reference, after renormalizing the Maslov gradings to agree with the link Floer homology, the knot We now show that L has the same link Floer Homology as T (2, 6), so it follows from Theorem 3.2 that Since the linking number is 3 and the Maslov index 0 generator sits in Alexander grading 0 in the knot Floer homology of each component, it follows that in HFL(L) the Maslov index 0 generator sits in Alexander bi-grading ( 3 2 , 3 2 ).The Maslov index −1 generators in HFK(L) must be in bi-Alexander gradings ).There is also Maslov index −2 generator in Alexander grading ( 6 Link Floer homology detects T (n, n) In the previous section we showed that link Floer homology detects the T (2, 2n) torus links, motivated by detection results for T (2, 2), T (2, 4), and T (2, 6).The torus link T (2, 2) can also be viewed as one of the simplest links in the family of T (n, n) torus links.In this section we show that link Floer homology detects the links T (n, n).We use a characterization of T (n + 1, n + 1) as an (n)-braid for T (n, n) union the braid axis.
In [18], J. Licata computes the link Floer homology of the links T (n, n), aside from the Maslov gradings of certain generators when n > 6.She conjectures a complete result.We prove that link Floer homology detects T (n, n) using her computation.It follows from this that there are many graded vector spaces that do not arise as the link Floer homology of any link.
We will be interested in multi-graded vector spaces B n exhibiting the following four properties: 1.There is a unique Maslov grading 0 generator.
3. B n has support contained only in multi-Alexander gradings (A 1 , A 2 , . . ., A n ) satisfying A i ≤ n−1 2 for 10 all i.
4. B n has rank 2 n−1 in A i grading n−1 2 .
In particular HFL(T (n, n)) satisfies the above properties.Observe that if L is any link whose link Floer homology satisfies all of the above conditions then L is not a split link, so each component The main ingredient of this proof is a result stating that, under certain circumstances, if the link Floer homology of a link has certain algebraic properties then the linking numbers of certain components with the rest of the link are positive.
Lemma 6.2.Let L be a link with components L i for 1 ≤ i ≤ n.Suppose that HFL(L) has a unique generator of Maslov index 0 with A i grading x ≥ 0. Suppose HFL(L) is supported in A i gradings at most x.Then ℓk(L i , L j ) ≥ 0 for all j.
Proof of Lemma 6.2.Let θ 0 denote the unique Maslov index 0 generator.The vector space HF(S 3 )⊗V n−1 is non-zero in Maslov grading 0 so all other intermediate vector spaces with spectral sequences fitting between HFL(L) and HF(S 3 ) ⊗ V n−1 must also be non-zero in this Maslov grading.Because θ 0 is the only generator in this Maslov grading, it must survive in every such spectral sequence.
Consider the spectral sequence to HFL(L − L j ) ⊗ V obtained by forgetting the component L j .The A i grading on HFL(L − L j ) ⊗ V will be shifted by ℓk(Li,Lj) 2 , we show that this shift must be non-negative.
Because θ 0 survives this spectral sequence, HFL(L − L j ) ⊗ V will have top A i grading n−1 2 .Considering the A i grading on HFL(L) we see that HFL(L − L j ) ⊗ V will have bottom A i grading no smaller than −n+1 2 .
Since the A i grading on HFL(L − L j ) ⊗ V must be symmetric about a non-negative number, the shift applied to the Alexander grading must be non-negative, so ℓk(L i , L j ) ≥ 0.
With this result on the non-negativity of linking numbers, we can proceed with the proof of Theorem 6.1.
We will proceed by induction, using the characterization of T (n+ 1, n+ 1) as the link consisting of the unique n-braid for T (n, n) together with the braid axis.
Proof of Theorem 6.1.Suppose that HFL(L) ∼ = HFL(T (n, n)).Lemma 6.2 tells us that ℓk(L i , L j ) ≥ 0 for every distinct i, j.Moreover, because L is not split and each component L i of L is a braid axis for L − L i we have ℓk(L i , L j ) = 0.
The top non-zero A i grading is n−1 2 .The relationship between the top non-zero A i grading and the Seifert genus of L i implies that; However, because ℓk(L i , L j ) > 0 we also have that with equality when ℓk(L i , L j ) = 1 for all j.Combining these inequalities gives that g(L i ) = 0, and ℓk(L i , L j ) = 1 for all i, j.
We now know that L is an n-component link where each component is an unknot, each component is a braid axis for the rest of the link, and the linking number between any two components is 1.The torus link T (n, n) is the only n component link satisfying all of these conditions.This can be verified by induction on n.Specifically, check explicitly that T (2, 2) is the only such 2-component link, then view L as a braid axis of some n-braid representing an n component link satisfying the same properties.
7 Knot Floer homology detects T (3, 3) In previous sections we showed that for some of the first members of the family of T (2, 2n) torus links the link Floer homology detection result can be strengthened to knot Floer homology detection results.In this section we do the same for T (3, 3), the third member of the T (n, n) family.To prove we will use various spectral sequence arguments to show that L has the same link Floer homology as T (3, 3).The above theorem then follows immediately from Theorem 6.1.
First note that n ≤ 3 as else HFK(L) would not admit a spectral sequence to HF(# n−1 (S 1 × S 2 )).n = 2 since the Maslov gradings of HFK(L) are supported in integer gradings.Moreover, L cannot be a knot since rank( HFK)(L) is even.Thus n = 3.
Let L 1 , L 2 , L 3 be the components of L. We now seek to determine the structure of HFL(L).
The symmetry of HFL(L) implies that the unique generator in Maslov grading −2 and Alexander grading 0 sits in multi-Alexander grading (0, 0, 0).Similarly the symmetry implies that at least one of the Maslov grading −3 generators also sits at multi-grading (0, 0, 0) .
Since the Maslov grading 0 generator in knot Floer homology is of Alexander grading, 3, the Maslov grading 0 generator in Link Floer homology sits in Alexander multi-grading (x, y, 3 − x − y) for some pair of integers (x, y).In order that the link Floer homology admits the requisite spectral sequences, there are Maslov By a similar argument, we can see that that there is a Maslov index −3 generator in multi-Alexander grading (x − 1, y − 1, 2 − x − y).If (x, y) = (1, 1) this determines the entire link Floer complex.If x = y = 1 then the remaining Maslov index −3 generators must be of multi-Alexander grading (0, 0, 0) to insure that each (A i , A j ) grading is of even rank, so again the entire complex is determined.
Since rank( HFK(L)) = 18, rank( HFK(L i )) ≤ 9 2 .It follows that each component is an unknot or a trefoil.Observe that if a component is a trefoil then it must be T (2, 3), as there are no positive Maslov index generators in HFK(L).Indeed there can be no T (2, 3) component as this would require there to be an Alexander grading two less than an Alexandar grading of the Maslov index 0 generator containing a summand F −2 ⊕ F 2 −3 ⊕ F −4 , which does not occur.Thus each component is an unknot.From here we can compute the linking numbers from the Alexander gradings of the Maslov grading 0 generator.We find in the complement of L 3 , when one of the two strands is given the opposite orientation.Each of L 1 and L 2 are unknots and ℓk(L 1 , L 2 ) = 3 so after changing the orientation of L 1 , L − L 3 is the 2-braid T (2, −6).However, the rank of knot Floer homology is invariant under changing orientations and rank( HFK(T (2, 6) ⊗ V )) = 24, so T (2, 6) cannot be a sublink of L and (x, y) = (1, 2).

Knot Floer homology detects L7n1
We have now shown that knot Floer homology detects a number of the low crossing number links that Khovanov homology is known to detect.In this section we continue this task, showing that knot Floer homology detects the link L7n1.
Our proof relies on the observation that L7n1 can be realized as a 2-braid representing T (2, 3) together with the braid axis.
First note that it follows from the fact that HFK(L) admits a spectral sequence to HF(# n−1 (S 1 × S 2 )) -where n is the number of components of L -and the fact that the knot Floer homology of a knot is of odd rank that L is a two component link.Since knot Floer homology detects the linking number of two component knots, it follows that the linking number of L is two.From here we break up the proof of The combination of these Lemmas immediately gives the proof of Theorem 8.1.
L7n1 has homology as computed in [25], and shown in Table 2.
Lemma 8.2 is proven by combining the symmetry and parity properties of the link Floer homology complex.
Proof of Lemma 8.2.Since L has two components, HFL(L) has exactly 2 Alexander gradings.
Let θ 0 be the Maslov grading 0 generator.This generator θ 0 has bi-Alexander grading ( 3 2 + x, 3 2 − x) for some x.Indeed, there must be generators sitting in gradings ( Together with the symmetry properties of link Floer homology, this determines the Alexander bi-gradings of 6.The same symmetry properties also imply that the two generators in Alexander grading 0 must have bi-Alexander grading (0, 0).Thus, up to choice of x, we need only specify the location of one more generator to determine the whole link Floer homology complex.Since each Alexander grading to be of even rank, the remaining Maslov grading −2 element must be in bi-Alexander grading ( 1 2 + x, 1 2 − x).Moreover, since the Maslov grading −3 component, θ −3 , cannot persist in the spectral sequence to HF(S 3 )⊗V , it follows that x ∈ { 1 2 , 0, − 1 2 }, for the Alexander gradings obstruct the existence of generators y with ∂y, θ −3 = 0, 15 and ∂θ −3 can only be non-zero if x is in this range.Indeed, x = 0, since otherwise we would have an element with Alexander grading in Z, and another with Alexander grading in Z + 1 2 .The remaining two possibilities give link Floer homologies that agree with L7n1, as desired.
We complete the proof of Theorem 8.1 by showing that link Floer homology detects the link L7n1.We use the fact that L7n1 is the closure of a braid for T (2, 3) together with its braid axis.
Observe that rank( HFK(L i )) ≤ 5, with equality if and only if the spectral sequence corresponding to L i collapses on the E 1 page.If the spectral sequence collapses on the E 1 page, then HFK(L i ) would have no shift applied to its Alexander grading as it is already symmetric around grading 0. Therefore, if rank( HFK(L i )) = 5 then ℓk(L 1 , L 2 ) = 0.However, this is impossible because L is not split and so the link Floer homology shows that L − L i is braided with respect to L i .
Thus rank( HFK(L i )) < 5 and the link has components that are either unknots and trefoils.Observe that any trefoil component must be T (2, 3), as there are no generators of positive Maslov grading.
Observe that L 1 cannot be a trefoil for there is no Maslov grading −2 generator in an A 1 grading two less than the A 1 grading of the unique grading 0 element.Thus L 2 is T (2, 3) and ℓk(L 2 , L 1 ) = 2. Since L 2 is a 2-braid closure in the complement of L 1 and L 2 is T (2, 3), the link L must be L7n1.

Connected sums with a Hopf link
In this section we deduce some properties of link Floer homology under the operation of taking a connected sums with a Hopf link.We then explore some applications of these properties to the question of link detection.10 Applications to annular Khovanov homology Annular Khovanov homology was defined by Asaeda-Przytycki-Sikora [1] as a categorification of the Kauffman bracket skein module of the thickened annulus.The resulting theory is an invariant of links in the thickened annulus A × I or alternatively the complement of an unknot in the 3-sphere S 3 \ U .In particular, annular Khovanov homology is well suited to studying braid closures [2,10,14,15].
In this section we apply some of our earlier knot Floer detection results to show that annular Khovanov homology detects certain braid closures.The proofs will rely on the spectral sequence from annular Khovanov homology of a link L to the knot Floer homology of the lift of the annular axis in the Σ(L) [10,28].
We With the previous lemmas we will complete the proof of Theorem 10.4 by understanding the symmetries of T (2, 4).
Proof of Theorem 10.4.The complement of the torus link T (2, 4) is a Seifert fibered space with two exceptional fibers and the base space is an annulus.Any symmetry of T (2, 4) is isotopic to a symmetry which preserves the fibration [5].The symmetries must also preserve the exceptional fibers so we can consider symmetries of the annulus with two marked points.There is an order two reflection that reverses the orientation of the annulus and an order two hyper elliptic involution that preserves orientation and interchanges the boundary components of the annulus.These two symmetries commute and generate the all symmetries of the annulus with two marked points.This shows that the symmetry group of T (2, 4) is Z/2Z ⊕ Z/2Z and Theorem 10.4 without considering the symmetries of T (2, 4).The same idea would be true for Theorem 10.7 and a similar question for the Birman-Hilden correspondence for B 6 .

Theorem 6 . 1 .Proposition 9 . 2 .
If HFL(L) ∼ = HFL(T (n, n)), then L is isotopic to T (n, n).If knot Floer homology detects a link L, then link Floer homology detects L#H for each choice of component of L to connect sum with.

[ 7 ]Lemma 3 . 3 . 1 2 , and 3 2
and also appears in B. Liu's classification of the links T (2, 2n) in terms of surgery to a Heegaard Floer L-space[19].If HFK(L) ∼ = HFK(J n )) for some n, then L is a 2-component link and both of the components are unknots.Proof of Lemma 3.3.First we show that L is a 2-component link.Notice that the parity of the rank of HFK(L) rules out the case that L is a knot.If L is an n-component link then there is a spectral sequence from HFK(L) to HF(# n−1 (S 1 × S 2 )).Because HFK(L) is only non-zero in Maslov gradings − 1 2 , this spectral sequence can only exist for n = 2.

Theorem 9 . 4 .Remark 9 . 5 . 6 .Theorem 9 . 6 . 3 .
Our main application is to find two infinite families of links which are not detected by Khovanov homology or knot Floer homology but which are detected by link Floer homology.Throughout this section we let H denote the Hopf link. the second family, let L (a,b) be the tree of unknots corresponding to the graph with a + b + 2 vertices with a vertices connected to a vertex x, b vertices connected to a vertex y and an edge connecting the vertex x and the vertex y.For each n ≥ 2, if HFL(L) ∼ = HFL(L n ), then L is isotopic to the link L n .The links L n can be viewed as the trivial n − 1-braid together with its braid axis.So Theorem 9.4 was already known because link Floer homology detects braid closures [20, Proposition 1] and detects the trivial braid amongst braid closures [2, Theorem 3.1].However, we provide a different proof of Theorem 9.4 because it is a simpler case of the ideas used in the proof of Theorem 9.Proof of Theorem 9.4.Suppose L has the same link Floer homology as L n .First notice that L cannot be a split link because there is a generator with all of its Alexander gradings non-zero.Additionally we see that n − 1 components of L each bound a disk intersecting the rest of L in exactly one point.By the observation that L is not split, we see that each of these n − 1 components must bound a disk which only intersects the final component of L. So then L is isotopic to the link L n .For every pair (a, b) with a and b positive, if HFL(L) ∼ = HFL(L (a,b) ), then L is isotopoic to the link L (a,b) .Proof.First notice that link Floer homology detects the link L (0,b) = L b+1 .We will now proceed by induction on a.Suppose that L has the same link Floer homology as L (a,b) .First notice that L cannot be a split link because there is a generator with all of its Alexander gradings non-zero.Additionally we see that a + b components of L each bound a disk intersecting the rest of L in exactly one point.By the observation that L is not split, we see that each of these a + b components must bound a disk which only intersects one of the final two components of L. Call these final components X and Y based on if their Alexander gradings agree with the Alexander gradings associated to the component in the tree of unknots for the vertex x or y respectively.Without loss of generality, at least one component bounds a disk that intersects X in a single point.Then L can be written as L ′ #H where the connect sum is taken along the component X.A quick computation shows that HFL(L ′ ) ∼ = HFL(L (a−1,b) ).By induction L ′ is isotopic to L (a−1,b) whence L is isotopic to L (a,b) .With these detection results in place, we are now ready to prove Theorem 9.Proof of Theorem 9.3.Consider the links L n and L (a,b) with a + b + 1 = n.These links are detected by link Floer homology.We now check that Kh(L n ) ∼ = Kh(L (a,b) ) and HFK(L n ) ∼ = HFK(L (a,b) ).Both links can be constructed by starting with an unknot and connect summing a Hopf link n times in total.A simple computation shows that Khovanov homology and knot Floer homology of L#H does not depend on which component of L the Hopf link is connect summed onto.This shows Kh(L n ) ∼ = Kh(L (a,b) ) and HFK(L n ) ∼ = HFK(L (a,b) ).

2 , σ 1 σ 2 σ 3 , and σ 1 σ 2 σ 3 σ 4 σ 5 .
use knot Floer detection results for T (2, 3), T (2, 4) and T (2, 6) to show annular Khovanov homology detects the closure of the braids σ 1 σ The structure of each proof is similar; first we use properties of annular Khovanov homology to deduce necessary topological properties of the annular knot like braidedness or unknottedness.Then we use a knot Floer detection result to show that the lift of the annular axis is T (2, 3), T (2, 4) or T (2, 6) respectively.Finally we translate this into information about the annular link.The spectral sequence from the annular Khovanov homology of an annular link L to the knot Floer homology of the lift of the annular axis in Σ(L) is defined with Z/2Z coefficients, at times however we will From the construction of the spectral sequence from AKh(L, Z/2Z) to HFK(− U ) as an iterated mapping cone, it follows that in each i grading on AKh(L, Z/2Z) the relative Maslov grading of any two generators agrees with half the difference of the quantum gradings of the generators.It remains to relate the relative Maslov gradings for generators in i grading 0 and 1 and then upgrade this information to an absolute Maslov grading.The induced differential ∂ − giving the spectral sequence from AKh(L, Z/2Z) to Kh(L, Z/2Z) is part of the total differential on the iterated mapping cone induced by counting pseudo holomorphic polygons.Thus ∂ − lowers the Maslov grading by one.This implies that for AKh(L, Z/2Z), generators in the same k grading live in the same relative Maslov grading.Since generators in the same k grading or 2A grading also have the same Maslov grading, the spectral sequence to HFK(− U ) collapses as all differentials preserve the k grading or 2A grading and change the Maslov grading.To upgrade the above to a statement about the absolute Maslov grading, notice that there are only two generators survive in the spectral sequence to Kh(L, Z/2Z), namely the generators that sit in the k gradings −4 and −2.These generators must then be in Maslov gradings 0 and 1 respectively.From here we can pin down the Maslov gradings of the remaining six generators of AKh(L, Z/2Z).The claim HFK( U ) ∼ = HFK(T (2, 4)) then follows from the fact that HFK(− U ) ∼ = ( HFK( U )) * with the appropriate change in gradings.An application of Theorem 4.1 completes the proof of Lemma 10.6.

Table 2 :
The link Floer homology of L7n1.