Annular Khovanov homology and augmented links

Given an annular link $L$, there is a corresponding augmented link $\widetilde{L}$ in $S^3$ obtained by adding a meridian unknot component to $L$. In this paper, we construct a spectral sequence with the second page isomorphic to the annular Khovanov homology of $L$ and it converges to the reduced Khovanov homology of $\widetilde{L}$. As an application, we classify all the links with the minimal rank of annular Khovanov homology. We also give a proof that annular Khovanov homology detects unlinks.


Introduction
Khovanov [Kho00] defined an invariant for links which assigns a bigraded abelian group Kh(L) for each link L ⊂ S 3 .It is a categorification of Jones polynomial in the sense that it replaces terms in Jones polynomial by graded abelian groups.Since then, many related invariants have been studied, including Lee's deformation and Rasmussen's s-invariant [Lee05,Ras10], the reduced version [Kho03], the thickened surface version [APS04], the tangle invariant [BN05] and Khovanov-Rozansky homology [KR08], etc.. Several spectral sequences that reveal the relationship between Khovanov homology theories and Floer theories have been established.The first one is due to Ozsváth and Szabó [OS05] that builds a connection between the reduced Khovanov homology of the mirror of a link L and the Heegaard Floer homology of the branched double cover of S 3 over L. Kronheimer and Mrowka [KM11] constructed a spectral sequence with the E 1 term isomorphic to Khovanov homology and converging to a version of singular instanton Floer homology.
Let A be an annulus (sometimes it is convenient to view A as a punctured disk).Then the theory of thickened surface [APS04] applies for A × I, which is called annular Khovanov homology.Roberts [Rob13] constructed a spectral sequence from annular Khovanov homology to Heegaard Floer homology.The analogue of Rasmussen's s-invariant in the annular settings was studied [GLW17].Xie [Xie21] introduced annular instanton Floer homology for annular links as an analogue of the annular Khovanov homology, and they are also related by a spectral sequence, which can be used to distinguish braids from other tangles [Xie21,XZ19b].
The relationship between annular Khovanov homology and the original Khovanov homology was studied.There is a natural spectral sequence between them given by ignoring the punctured point [Rob13, Lemma 2.3].However, considering the augmentation of links is more helpful to preserve the information about the punctured point.
Definition 1.1.Let L ⊂ A × I be an annular link.The augmentation of L is a pointed link ( L, p) ⊂ R 3 obtained as follows.We view the thickened annulus A × I as a solid torus in R 3 , and L is given by the union of L and a meridian circle of A (sometimes we call it an augmenting circle).The base point p is chosen on the augmenting circle.Under this convention, Xie [Xie21, Section 4.3] showed that the annular instanton Floer homology AHI(L) is isomorphic to I ♮ ( L), the reduced singular instanton Floer homology of the augmented link.In this paper, we prove the following theorem as an analogue of Xie's result in the Khovanov side.To avoid the sign issues, all the coefficient rings will be Z/2Z unless otherwise specified.
Theorem 1.2.Let L ⊂ A × I be an annular link and let ( L, p) ⊂ S 3 be the corresponded augmented link of L. Then there is a spectral sequence with the E 2 term isomorphic to the annular Khovanov homology AKh(L) and it converges to the reduced Khovanov homology Khr( L, p).
We immediately obtain the following rank inequality.
Corollary 1.3.Given an annular link L and its augmentation L, we have Question 1.4.For what link L we have AKh(L) is isomorphic to Khr( L, p)? Theorem 1.2 provides an alternative way to prove some detection results by referring to the parallel consequences in reduced Khovanov homology.For a link L with n components, it is well-known that rank Z/2Z Khr(L, p) ≥ 2 n−1 .Hence by the previous corollary, for an annular link L, we have On the other hand, links of minimal rank in A × I can be classified following [XZ19a].Before state the result, we first explain the notation.Given a forest G, its corresponding link L G is defined by assigning each vertex of G an unknot component and linking two unknots in the way of Hopf links whenever their corresponding vertices are adjacent.For annular links, the only additional rule is that we need to assign which vertex is corresponding to a nontrivial circle.We say such vertices are annular for convenience.
Theorem 1.5.Let L be an n-component annular link.Then rank Z/2Z AKh(L) = 2 n if and only if L is a forest of unknots such that each connected component of the corresponding graph of L contains at most one annular vertex.
We say an annular link U is an unlink if it has a link diagram D without any crossing.Notice that our definition given here is slightly different to [Xie21].The following result is a generalization of [XZ19b, Corollary 1.4].
Corollary 1.6.Let L be an annular link with n components and let U be an annular unlink with n components (might be trivial or nontrivial).Assume that as bigraded (by homological and Alexander gradings) abelian groups.Then L is isotopic to U .
The paper is organized as follows.In Section 2 we review the construction and properties of Khovanov homology.After some preparation in Section 3, we prove Theorem 1.2 in the last section and discuss its applications.
Acknowledgement.The author would like to thank his advisor Yi Xie for introducing this problem to him and giving him patient and accurate guidance.The author is also grateful to Qing Lan and Xiangqian Yang for helpful conversations.This paper is part of the author's undergraduate research and is partially supported by the elite undergraduate training program of School of Mathematical Sciences, Peking University.

Review on Khovanov homology theories
In this section, we review the construction and properties of the reduced version and the annular version of Khovanov homology.In the case of original Khovanov homology, we apply a (1 + 1)d TQFT to the resolution cube to obtain a chain complex by assigning each circle a graded free abelian group V := Z/2Z{v + , v − }.The resulted complex has two gradings: the homological one and the quantum one, and the latter is specified by q deg v ± = ±1.Following [BN02], we denote the shift on these two gradings by [•] and {•}, respectively.We then take a shift on the quantum grading of chain groups by |v| to ensure the differential preserves the quantum grading and a global shift [−n − ]{n + − 2n − } to ensure the invariance under Reidemeister moves.We finally take cohomology on the chain complex (CKh(L), d) to obtain Kh(L).
To define the reduced version of Khovanov homology, as in other reduced theories, we need to choose a base point p on the link L. Every resolution of L has exactly one circle containing p, and the generators that take v − (with the qgrading omitted) on this pointed circle span a subcomplex CKhr(L, p) ⊂ CKh(L).The reduced Khovanov homology Khr(L, p) is then defined by the cohomology of CKhr(L, p).The base point is sometime omitted in the notation if it is clear from the text (e.g. when we are considering an augmented link).As an example, for Hopf link H with a positive linking number, we have Khr(H, p) = (Z/2Z) (0,1) ⊕ (Z/2Z) (2,5) .
In general, the following proposition describe the effect on Khovanov homology of making a connected sum with a Hopf link.
Proposition 2.1 ([AP04, Theorem 6.1]).Let L be a pointed link and let H be the Hopf link with a positive linking number.Then we have a short exact sequence: Here α * and β * are given on a state S as in Figure 3.
2.2.Annular Khovanov homology.The annular version of Khovanov homology can be viewed as a special case of the link homology for links in thickened surfaces defined in [APS04].Let A be an annulus.The annular Khovanov homology assigns a triply-graded abelian group AKh(L) for each annular link L ⊂ A × I.We follow the process and notation of [Xie21].
Let D be a link diagram of L and define n, n ± , v, D v , V as in the previous subsection.In the annular case, there might be two types of circles in a resolution: circles that bound disks and circles with nontrivial homologies.We call the first type of circles trivial and the second ones nontrivial.To obtain the chain groups, we assign V to trivial circles and assign W := Z/2Z{w + , w − } to nontrivial circles.The differentials are specified by the map corresponding to the merging or splitting of circles, as follows.
• Two trivial circles merge into a trivial circle, or one trivial circle splits into two trivial circles.In these cases, the maps are given as same as in Khovanov's original TQFT.• One trivial circle and one nontrivial circle merge into a nontrivial circle.In this case, the maps are given by • One nontrivial circle splits into a trivial circle and a nontrivial circle.In this case, the maps are given by • Two nontrivial circles merge into a trivial circle.In this case, the maps are given by • One trivial circle splits into two nontrivial circles.In this case, the maps are given by The homological and quantum grading are given as same as the original case with the additional request that q deg w ± = ±1.After appropriate shifts, the differential is still filtered of degree (1, 0).
There is the third grading on the chain complex, so-called the Alexander grading or f -grading, which is specified by f deg v ± = 0 and f deg w ± = ±1.The differential preserves the f -grading and hence it descends onto the cohomology groups AKh(L), the annular Khovanov homology.

Theorem 2.2 ([APS04]). The annular Khovanov homology AKh(L) is an invariant of links in the sense that it is independent of the choice of link diagrams and the order of crossings.
We conclude this section by some additional remarks.Sometimes we write AKh(L, m) to indicate the f -degree m summand of AKh(L).If L is contained in a ball B 3 ⊂ A × I, then AKh(L) is supported on f = 0 and AKh(L) ∼ = Kh(L).Both the reduced Khovanov homology and the annular Khovanov homology are functorial.That is, a cobordism ρ : L 1 → L 2 between links (resp.annular links) induces a (filtered) map between Khovanov homology groups Khr(ρ) : Khr(L 1 ) → Khr(L 2 ) (resp.AKh(ρ) : AKh(L 1 ) → AKh(L 2 )).

The unlink case
In this section, we construct an isomorphism between the annular Khovanov homology of an annular unlink and the reduced Khovanov homology of its augmentation.We show that such an isomorphism is compatible with the group homomorphisms induced by the cobordism maps.
3.1.Homology groups.Denote the annular unlink with n nontrivial unknot components by U n and let U n be its augmentation.In the language of [XZ19a], U n corresponds to the graph shown in Figure 4.The obvious diagram of U n contains n disjoint nontrivial circles.In this section, we will stick on this diagram to calculate homology groups.We assign the number 1 to n from the innermost nontrivial circle to the outermost one.By Proposition 2.1, the Poincaré polynomial of Khr( U n ) is given by Here the homological and quantum grading are indicated by t and q respectively.Each original component of U n has two crossings with the meridian circle.There are 2 n resolutions such that every pair of crossings is resolved by the same smoothing.We say such resolutions are symmetric and encode them by 0 − 1 sequences of length n, as shown in Figure 5. Notice that a symmetric resolution always has n (unpointed) components.We denote the cobordism of changing one crossing (on the k-th strand) from 0-smoothing to 1-smoothing by ( We can now describe the generators of Khr( U n ) explicitly.
Proposition 3.1.For each symmetric resolution v ∈ {0, 1} n , we can choose an element e v lying in the chain group corresponding to this resolution.The collection of e v descends to a generating set of Khr( U n ).
Proof.We prove the proposition by induction.There is nothing to say for n = 0.In general, by applying proposition 2.1 to L = U n−1 and L#H = U n , we obtain a short exact sequence and it remains to show that e v is a cycle.Notice that the cobordism (v ′ , •) is always a merging (rather than a splitting) of circles, and the construction ensures that Khr((v ′ , •))(e v ) = 0. We show that other cobordisms also vanish by discussing the value of v n−1 , see Figure 6.Notice that the cobordism map that the change happens on the i-th strand (1 ≤ i ≤ n − 2) vanishes on A, B. Hence if v n−1 = 1, then there is no possibly nonvanishing cobordism map.Now assume that v n−1 = 0 and let v ′′ = (v 1 , . . ., v n−2 ),

Then we have
We now construct an explicit identification between AKh(U n ) and Khr( U n ).On the level of homology, this is quite easy: the Poincaré polynomial of AKh(U n ) is given by Here the f -grading is indicated by f .The substitution f → tq gives an isomorphism between AKh(U n ) and Khr(U ′ n ) (up to shifting).More concretely, the generator w = w (1) is identified with the generator corresponding to the symmetric resolution of label (v 1 , v 2 , . . ., v n ), where v i = 1 if and only if w The effect of adding a trivial unknot component to U n is just taking two copies of the original homology groups with generators tensoring with v ± respectively, by Künneth formula.We summarize the consequence of this subsection in the following form.
Theorem 3.2.Let L be an annular unlink with n nontrivial unknot components, and let L be its augmentation.Then there is an isomorphism Φ L between the annular Khovanov homology of L and the reduced Khovanov homology of L.More precisely, we have an isomorphism The correspondence of generators is given above.

Functority.
A cobordism between annular links naturally induces a cobordism between their augmentations.In this subsection, we show that the isomorphism Φ L defined in Theorem 3.2 is compatible with cobordisms.According to our purpose (see the next section), we don't need to deal with the Reidemeister moves on the diagram of L, and we concentrate on Morse moves, i.e. the merging and splitting of circles.We first verify the compatibility with only related circles and then consider the effect of adding other unlink components.There are four cases we need to discuss: a. one trivial circle and one nontrivial circle merge into a nontrivial circle; b. one nontrivial circle splits into a trivial circle and a nontrivial circle; c. two nontrivial circles merge into a trivial circle; d. one trivial circle splits into two nontrivial circles.
This completes the verification in case d..It remains to consider the effect of adding a new unlink component to the cobordism.The case of adding a trivial unknot component is trivial and we assume that the additional unknot component is nontrivial.Let L 1 , L 2 be two annular unlinks and let ρ : L 1 → L 2 be a cobordism obtained by a Morse move.We have AKh(ρ id) = AKh(ρ) ⊗ id U , here U = U 1 is the nontrivial annular unknot.Take S ∈ AKh(L 1 ) and let T = AKh(ρ)(S).By Proposition 2.1 and Theorem 3.2, the following diagram commutes.
In summary, we have shown the following theorem.Roughly speaking, it gives a natural isomorphism between two cohomology theories on annular unlinks.

Figure 1 .
Figure 1.An annular link and its augmentation.

Figure 4 .
Figure 4.The tree corresponding to U n .

v − ⊗ v + 0 Figure 7 .Figure 8 .
Figure 7. Case a..In the case c. and d., we need to check the following diagrams commute.Φ L3

Figure 9 .
Figure 9.The label of crossings on L 4 .