KKHOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS
DAVID BOOZER
Abstract.
We show that the reduced Khovanov homology of a link L in S can be expressed as thehomology of a chain complex constructed from a description of L as the closure of a 1-tangle diagram in theannulus. The chain complex is constructed using a resolution of the 1-tangle diagram into planar tangles ina manner analogous to ordinary reduced Khovanov homology, but with some novel features. In particular,unlike for ordinary Khovanov homology, terms appear in the differential that are products of linear mapscorresponding to pairs of saddles obtained from the resolution. We also use the chain complex to constructa spectral sequence that converges to reduced Khovanov homology. Introduction
Khovanov homology is a powerful invariant defined for oriented links in S [4]. The Khovanov homologyof an oriented link is a finitely-generated bigraded abelian group that categorifies its Jones polynomial [3];roughly speaking, the relationship between the Khovanov homology of a link and its Jones polynomial isanalogous to the relationship between the singular homology of a topological space and its Euler charac-teristic. The Jones polynomial of a link can be recovered from its Khovanov homology, but the Khovanovhomology generally contains more information: it can sometimes distinguish between links with the sameJones polynomial, and Khovanov homology detects the unknot [5], but it is not known whether the same istrue of the Jones polynomial.Here we formulate reduced Khovanov homology in terms of descriptions of links as closures of 1-tanglediagrams in the annulus. A is a 1-dimensional submanifold of A × I , where A = S × I is the annulus,with boundary { ( p, / , ( q, / } , where p and q are fixed points on the inner and outer boundary circlesof A . A 1-tangle can be projected onto the annulus A to yield a . For a generic 1-tangle,the associated 1-tangle diagram is an immersed 1-manifold with finitely many transverse double-points withcrossing data.Given an oriented 1-tangle diagram T , we can close T using an unknotted overpass or underpass arc toobtain oriented link diagrams T + and T − . These link diagrams determine links in S that are defined upto isotopy, which for simplicity we also denote by T + and T − . We construct a bigraded chain complex( C T ± , ∂ T ± ) by resolving the tangle diagram T into planar tangles in the annulus. We prove: Theorem 1.1.
The chain complex ( C T ± , ∂ T ± ) is chain-homotopy equivalent to the chain complex for thereduced Khovanov homology of the link T ± . In particular, the homology of the chain complex ( C T ± , ∂ T ± ) is the reduced Khovanov homology of thelink T ± .We also use the chain complex ( C T ± , ∂ T ± ) to construct a spectral sequence to reduced Khovanov homology.We can express the differential ∂ T ± as ∂ T ± = ∂ T ± + ∂ ± T ± , where ( ∂ T ± ) = 0. The term ∂ T ± is a sum of linearmaps corresponding to saddles obtained from the resolution of T , and the term ∂ ± T ± is a sum of products oflinear maps corresponding to pairs of saddles obtained from the resolution of T . We prove: Theorem 1.2.
There is a spectral sequence with E page given by the homology of ( C T ± , ∂ T ± ) that convergesto the reduced Khovanov homology of the link T ± . If we forget the bigrading of the chain complex ( C T ± , ∂ T ± ), we have that C T + = C T − as ungraded vectorspaces and ∂ T + = ∂ T − as linear maps of ungraded vector spaces. So the E pages of the spectral sequencesfor the links T + and T − are the same up to grading shifts.Khovanov homology is currently defined only for links in S , and an important open problem is togeneralize Khovanov homology to links in arbitrary 3-manifolds. The results presented here are part of an Date : February 23, 2021. a r X i v : . [ m a t h . G T ] F e b DAVID BOOZER attempt to construct Khovanov homology for links in lens spaces. Our strategy for constructing Khovanovhomology for such links relies on first generalizing a result due to Hedden, Herald, Hogancamp, and Kirk.In [2], they show that Bar Natan’s functor from the tangle cobordism category for 2-tangles in the 3-ballto chain complexes can be factored through the twisted Fukaya category for a symplectic manifold R ( S , S along a 2-sphere thattransversely intersects the link in four points. To the 4-punctured 2-sphere they associate its traceless SU (2)character variety, which is the pillowcase R ( S , R ( T , SU (2) character variety for the 2-punctured2-torus. If so, the corresponding functor would provide a natural way of generalizing Khovanov homologyto links in lens spaces, since a lens space containing a link can always be Heegaard-split into a pair of solidtori glued along a 2-torus that transversely intersects the link in two points. Using results from [1], we wereable to describe some of the structure of the Fukaya category of R ( T ,
2) that is relevant to generalizingHedden, Herald, Hogancamp, and Kirk’s factorization result, assuming such a generalization is possible. Thisinformation, together with the results of [2], provided the clues needed to construct the chain complex forannular 1-tangle diagrams described here. The information we have obtained regarding the Fukaya categoryof R ( T ,
2) suggests that the factorization result of [2] does indeed generalize, but we have not proven this.Our hope is that further investigation of this Fukaya category may yield a candidate construction of reducedKhovanov homology for links in lens spaces, and perhaps a geometric interpretation of the reduced Khovanovhomology for (1 , C T ± , ∂ T ± ) froman oriented annular 1-tangle diagram T , and we sketch the proof of Theorem 1.1. Our proof of Theorem1.1 involves first reformulating the chain complex ( C T ± , ∂ T ± ) in terms of planar m -tangles in the disk. InSection 4, we discuss the relevant concepts involving such planar disk tangles. In Section 5, we reformulatethe chain complex ( C T ± , ∂ T ± ) in terms of planar disk tangles. In Sections 6, 7, and 8, we describe the planardisk tangles and saddles that are obtained from resolving the (nonplanar) annular 1-tangle diagram T . InSections 9 and 10 we combine the results of Sections 4–8 to complete the proof of Theorem 1.1. In Section11, we prove Theorem 1.2. In Section 12, we illustrate our results with an example.2. Vector spaces and linear maps
In this section we define several vector spaces and linear maps that we will use to construct the chaincomplex ( C T ± , ∂ T ± ). Our notation is generally consistent with that of [2].We define F to be the field of two elements. We also view F as a Z -graded F -vector space lying entirelyin grading 0. We define a Z -graded F -vector space A = F ·{ (1) A , x ( − } , where the superscripts indicate thatthe vectors 1 A and x are homogeneous with gradings 1 and −
1. We refer to the Z -grading on F and A as a quantum grading . We define the following F -linear maps: η : F → A, η (1) = 1 A , ˙ η : F → A, ˙ η (1) = x,(cid:15) : A → F , (cid:15) (1 A ) = 0 , (cid:15) ( x ) = 1 , ˙ (cid:15) : A → F , ˙ (cid:15) (1 A ) = 1 , ˙ (cid:15) ( x ) = 0 , x : A → A, x (1 A ) = x, x ( x ) = 0 , ∆ : A → A ⊗ A, ∆(1 A ) = 1 A ⊗ x + x ⊗ A , ∆( x ) = x ⊗ x,m : A ⊗ A → A, m (1 A ⊗ A ) = 1 A , m (1 A ⊗ x ) = m ( x ⊗ A ) = x, m ( x ⊗ x ) = 0 . The quantum gradings of these maps are η (1) , ˙ η ( − , (cid:15) (1) , ˙ (cid:15) ( − , (1 x ) ( − , ∆ ( − , m ( − . The graded vector space A , together with the multiplication m , comultiplication ∆, unit η , and counit (cid:15) ,gives Khovanov’s Frobenius algebra. For notational simplicity, given an F -vector space V we often identify V with F ⊗ V . For example, we have that m = ˙ (cid:15) ⊗ A + (cid:15) ⊗ x , ∆ = ˙ η ⊗ A + η ⊗ x . HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 3
The chain complex ( C T ± , ∂ T ± ) is built out of F -vector spaces that carry both a homological Z -grading h and a quantum Z -grading q . We will view the vector spaces F and A as lying entirely in homologicalgrading 0. Given a bigraded vector space V , we indicate the bigrading of a homogeneous vector v ∈ V with superscripts as v ( h,q ) . We define the vector space V [ h, q ] to be V with gradings shifted upwards by( h, q ). That is, if v ∈ V is homogeneous with bigrading ( h v , q v ), then the corresponding vector v ∈ V [ h, q ] ishomogeneous with bigrading ( h + h v , q + q v ).3. Chain complex for tangles in the annulus
Given an oriented 1-tangle diagram T in the annulus, we close T above or below with an unknotted arc toobtain oriented link diagrams T + and T − . For each link diagram T ± , we define a chain complex ( C T ± , ∂ T ± )as follows.First we describe the bigraded vector space C T ± . Let c denote the total number of crossings of T . Choosean arbitrary ordering of the crossings. Define the 0-resolution (1-resolution) of a crossing such that theoverpass turns left (right). Define I = { , } c to be the set of binary strings of length c . Given a binarystring i ∈ I , define T i to be the planar 1-tangle in the annulus obtained by resolving the crossings of T inthe manner indicated by i ; specifically, resolve crossing number k as indicated by the k -th bit of i .A planar 1-tangle in the annulus consists of a single arc connecting the marked points on the inner andouter boundary circles, together with a finite number of circle components. We orient the arc of each planar1-tangle in a direction from the inner boundary circle to the outer boundary circle; note that this orientationis unrelated to the orientation of the (nonplanar) tangle T . We leave the circle components unoriented.For each integer n , define a bigraded vector space˜ C n = (cid:77) { A ⊗ c ( T i ) [ r ( i ) , r ( i )] | i ∈ I such that w ( T i ) = n } , where r ( i ) is the number of 1’s in the binary string i ∈ I , c ( T i ) is the number of circle components ofthe planar tangle T i , and w ( T i ) ∈ Z is the number of times the (oriented) arc component of T i windscounterclockwise around the annulus. Recall that we define the link diagram T + , respectively T − , by closingthe tangle diagram T with an arc a that passes over, respectively under, the tangle diagram T . Definethe loop number (cid:96) of the pair ( T, a ) to be the total number of times a crosses T . We can depict the pair( T, a ) as shown in Figure 1; note that we assume all the crossings of T take place in a disk T D that doesnot intersect the overpass/underpass arc a . Define n + ( T ± ) and n − ( T ± ) to be the number of positive andnegative crossings for the link diagram T ± . For each integer n , define a grading-shifted vector space C n by C n = ˜ C n [ h ± ( T ) + h ± ( A n ) , q ± ( T ) + q ± ( A n )] , where h ± ( T ) = − n − ( T ± ) , q ± ( T ) = n + ( T ± ) − n − ( T ± ) ,h ± ( A n ) = (1 / (cid:96) ± n ) , q ± ( A n ) = (1 / (cid:96) ± n ) . Define bigraded vector spaces C T + and C T − by C T ± = (cid:77) n C n . We note that the grading shift ( h ± ( T ) , q ± ( T )) of C n is familiar from the definition of ordinary reducedKhovanov homology, with the convention that the reduced Khovanov homology of the unknot is F [0 , h ± ( A n ) , q ± ( A n )) will be explained in Lemma 6.7.Next we describe the differential ∂ T ± . We first define several linear maps by summing over saddles p → p (cid:48) obtained from the resolution of T . Define linear maps ∂ n : C n → C n by summing the following terms:(1) For a saddle p → p (cid:48) with w ( p ) = w ( p (cid:48) ) = n splitting a circle from the arc:˙ η ⊗ A ⊗ r : A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r + 1 . (2) For a saddle p → p (cid:48) with w ( p ) = w ( p (cid:48) ) = n merging a circle with the arc:˙ (cid:15) ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r. (3) For a saddle p → p (cid:48) with w ( p ) = w ( p (cid:48) ) = n splitting a circle from a circle:∆ ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ A ⊗ A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r + 2 . DAVID BOOZER (4) For a saddle p → p with w ( p ) = w ( p (cid:48) ) = n merging two circles: m ⊗ A ⊗ r : A ⊗ A ⊗ A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r + 2 , c ( p (cid:48) ) = r + 1 . Define linear maps ˜ ∂ Ln : C n → C n (respectively ˜ ∂ Rn : C n → C n ) by summing the following terms:(1) For a saddle p → p (cid:48) with w ( p ) = w ( p (cid:48) ) = n splitting a circle from the arc that connects to the arcon the left (respectively right) side: η ⊗ A ⊗ r : A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r + 1 . (2) For a saddle p → p (cid:48) with w ( p ) = w ( p (cid:48) ) = n merging a circle with the arc that connects to the arc onthe left (respectively right) side: (cid:15) ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r. Define linear maps Q n : C n → C n − by summing the following terms:(1) For a saddle p → p (cid:48) with w ( p ) = n and w ( p (cid:48) ) = n − A ⊗ r : A ⊗ r → A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r. Define linear maps P n : C n → C n +2 by summing the following terms:(1) For a saddle p → p (cid:48) with w ( p ) = n and w ( p (cid:48) ) = n + 2,1 A ⊗ r : A ⊗ r → A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r. The grading shifts in the definition of the vector spaces C n imply that the bigradings of the linear maps ∂ n ,˜ ∂ Ln , and ˜ ∂ Rn are ( ∂ n ) (1 , , ( ˜ ∂ Ln ) (1 , , ( ˜ ∂ Rn ) (1 , . The bigradings of the linear maps Q n and P n are different for T + and T − . For T + , the bigradings are( Q n ) (0 , − , ( P n ) (2 , . For T − , the bigradings are ( Q n ) (2 , , ( P n ) (0 , − . Define a linear map ∂ T ± : C T ± → C T ± with bigrading ( ∂ T ± ) (1 , by ∂ T ± = (cid:88) n ∂ n . Define linear maps ∂ ± T ± : C T ± → C T ± with bigrading ( ∂ ± T ± ) (1 , by ∂ + T + = (cid:88) n ( ˜ ∂ Ln − Q n + Q n ˜ ∂ Ln ) , ∂ − T − = (cid:88) n ( ˜ ∂ Ln +2 P n + P n ˜ ∂ Ln ) . Define linear maps ∂ T ± : C T ± → C T ± by ∂ T ± = ∂ T ± + ∂ ± T ± .Having defined the bigraded vector space C T ± and differential ∂ T ± , we are now ready to restate Theorem1.1 from the Introduction: Theorem 3.1.
Given an oriented 1-tangle diagram T in the annulus, the pair ( C T ± , ∂ T ± ) is a chain complexthat is chain-homotopy equivalent to the chain complex for the reduced Khovanov homology of the link T ± ,where the marked point for the reduced Khovanov homology is taken to be the endpoint of T that lies on theinner boundary circle of the annulus. Our basic tool for constructing the chain-homotopy equivalence is the following lemma:
Lemma 3.2 (Reduction Lemma) . Consider a chain complex ( C, ∂ ) such that the vector space C has theform C = A ⊕ B ⊕ B for vector spaces A and B and the differential ∂ : C → C has the form ∂ = ∂ A β β α ∂ B α B ∂ B . Then ( A, ∂ A + β α ) is a chain complex that is chain-homotopy equivalent to ( C, ∂ ) . HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 5
Figure 1.
An annular 1-tangle diagram T with loop number (cid:96) determines an annular1-tangle diagram T with loop number (cid:96) −
1. The circle labeled T D is the disk m -tanglecorresponding to T , where m = (cid:96) + 1, as explained in Section 4.1. The red curve labeled a is the overpass/underpass arc used to define the link diagrams T ± and T ± . Proof.
The fact that ∂ = 0 implies that ( ∂ A + β α ) = 0, so ( A, ∂ A + β α ) is a chain complex. Definelinear maps F : C → A and G : A → C by F = (cid:0) A β (cid:1) , G = A α . The fact that ∂ = 0 implies that F and G are chain maps. Define a linear map H : C → C by H = B . We have that GF = 1 C + ∂H + H∂, F G = 1 A , so F and G are chain-homotopy equivalences. (cid:3) We refer to the chain complex (
A, ∂ A + β α ) as the reduced chain complex corresponding to ( C, ∂ ), andsay that it is obtained by reducing on the identity map 1 B : B → B in ∂ . In the differential ∂ A + β α forthe reduced chain complex, we refer to terms of ∂ A as residual terms and terms of β α as reduction terms .We refer to the maps α and β as reduction factors .We can now sketch the outline of the proof of Theorem 3.1: Proof of Theorem 3.1.
We will prove the claim for the case T + ; the case T − is similar. We prove the claimby induction on the loop number (cid:96) of the annular 1-tangle diagram T . For the base case (cid:96) = 0, the chaincomplex ( C T + , ∂ T + ) is the chain complex for the reduced Khovanov homology of T + , so the claim is triviallytrue.For the induction step, assume the claim is true for all annular 1-tangle diagrams of loop number (cid:96) − T with loop number (cid:96) . As shown in Figure 1, we can flip theoutermost loop of T under the annulus to obtain an annular 1-tangle diagram T with loop number (cid:96) − T + and T + describe isotopic links in S , the chain complexes for the reducedKhovanov homology of T + and T + are chain-homotopy equivalent. The induction hypothesis implies that( C T + , ∂ T + ) is chain-homotopy equivalent to the reduced Khovanov homology of T + , and is hence chain-homotopy equivalent to the reduced Khovanov homology of T + . As we will describe in Section 6, we canapply the Reduction Lemma 3.2 to ( C T + , ∂ T + ) to obtain a reduced chain complex ( C red , ∂ red ), whose precisedefinition is given in Definition 6.4. To complete the proof, we will show that ( C red , ∂ red ) = ( C T + , ∂ T + ). InLemma 6.8 we show that C red = C T + as bigraded vector spaces, and in Lemmas 9.3 and 10.4 we show that ∂ red = ∂ T + . (cid:3) DAVID BOOZER
Figure 2.
A link diagram L with at least one crossing can be decomposed into two disk2-tangle diagrams D and D , or into two annular 1-tangle diagrams A and A . Remark 3.3.
A link diagram L in R with at least one crossing can be decomposed into two disk 2-tanglediagrams D and D or into two annular 1-tangle diagrams A and A , as shown in Figure 2. We have shownthat the factorization result of [2] can be generalized to the case of such disk 2-tangle diagram decompositions,and the resulting chain complex is as described by Theorem 3.1 for the case of annular 1-tangle diagramswith loop number (cid:96) = 1. 4. Disk tangles and saddles
In order to prove Theorem 3.1, we will first reformulate the chain complex ( C T ± , ∂ T ± ) in terms of aresolution of the oriented annular 1-tangle diagram T into planar tangles in the disk. In this section wediscuss the relevant concepts involving the planar disk tangles and saddles obtained from the resolution.4.1. Tangle diagrams in the disk.
Consider an oriented 1-tangle diagram T in the annulus and a choiceof overpass/underpass arc a used to define the link diagrams T + and T − . Recall that we defined the loopnumber (cid:96) of the pair ( T, a ) to be the number of times a crosses T . Given a pair ( T, a ), we can cut T alongthe arc a to obtain an oriented m -tangle diagram T D in a topological disk, where m = (cid:96) + 1. The boundaryof T D defines 2 m marked points on the boundary circle of the disk. Conversely, an m -tangle diagram T D in the disk determines a 1-tangle diagram T in the annulus and an overpass/underpass arc a , as shown inFigure 1, such that the pair ( T, a ) has loop number (cid:96) = m −
1. We enumerate the 2 m marked points on theboundary circle of the disk by counting counterclockwise around the circle, where points 1 and m + 1 arelocated at the top and bottom.4.2. Planar tangles in the disk.
We define d m to be the set of planar m -tangles in the disk. A planartangle p ∈ d m consists of m arcs that connect the 2 m marked points in pairs, together with some number c d ( p ) of circle components that we refer to as disk circles . A planar m -tangle p ∈ d m in the disk uniquelydetermines a planar 1-tangle p A in the annulus, which consists of a single arc together with some numberof circle components. We refer to an arc a of p ∈ d m as a strand arc if it lies on the single arc of p A anda circle arc if it lies on a circle component of p A . We refer to the set of circle arcs of p that lie on a givencircle component of p A as an annular circle , and we define c a ( p ) to be the number of annular circles of p .We define the circle number c ( p ) = c d ( p ) + c a ( p ) of p to be the number of circle components of p A . We referto the set of strand arcs of p as the strand component of p . We refer to the annular circles and disk circles of p as circle components of p . Given a planar tangle p ∈ d m with m ≥
2, we define arcs A , B , D , and C thatcontain the marked points 1, m , m + 1, and m + 2, respectively, as shown in Figure 3. Note that B (cid:54) = C ,but otherwise the arcs A , B , C , and D need not be distinct.We orient the strand arcs of a planar disk tangle p ∈ d m by orienting the single arc of the correspondingannular 1-tangle p A in a direction from the inner boundary circle of the annulus to the outer boundary circle.We leave the circle arcs of p unoriented. Our choice of orientation of the single arc of p A induces an orderingof the strand arcs of p , and we will use notation such as a < a and a ≤ a to indicate how strand arcs a and a of p are related under this ordering. To indicate that a sequence of circle arcs ( a , · · · , a n ) belongto the same annular circle and that the arcs are encountered in sequence as we move either clockwise or HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 7
Figure 3.
Arcs A , B , C , and D of a planar disk m -tangle p ∈ d m (we depict only aportion of each arc).counterclockwise starting from a , we use the notation a − a − · · · − a n ; equivalently a n − · · · − a − a .To indicate that a sequence of arcs ( a , · · · , a n ) belong to different annular circles, we use the notation a , a , · · · , a n .We define the winding number w ( p ) ∈ {− (cid:96), · · · , (cid:96) } of p to be the number of times the single arc of p A winds counterclockwise around the annulus. We will sometimes indicate the winding number and circlenumber of a planar tangle p by using subscripts and parentheses: p n ( r ) indicates that p has winding number w ( p ) = n and circle number c ( p ) = r .Given a planar tangle p ∈ d m , we assign a type τ ( a ) ∈ { e, u, c } to each arc a in p as follows. Startingat marked point 1, we move along the single arc of p A in the direction of its orientation (from the innerboundary circle of the annulus to the outer boundary circle) and alternately label each arc a of p that weencounter as earringed ( τ ( a ) = e ) or unearringed ( τ ( a ) = u ); this terminology is adapted from [2]. Note inparticular that τ ( A ) = e , τ ( D ) = e for m odd, and τ ( D ) = u for m even. This procedure assigns types toall of the strand arcs. Each remaining arc a is a circle arc, and we define its type to be τ ( a ) = c . We definethe type of a disk circle d to be τ ( d ) = c . We define e = u and u = e .The arc components of a planar tangle p ∈ d m describe a crossingless matching of the 2 m marked pointson the boundary of the disk. We define a set D m of such crossingless matchings; note that | D m | = C m ,where m is the m -th Catalan number. Given a planar tangle p ∈ d m , we define the corresponding crossinglessmatching τ ( p ) ∈ D m to be the type of the planar tangle. We can also think of τ ( p ) as the planar tangleobtained from p by removing all of the disk circles, and under this interpretation many of our notionsinvolving planar tangles carry over to tangle types. For example, we define the winding number w ( P ) of atangle type P to be the winding number of any planar tangle p of type τ ( p ) = P . We can also speak of thearcs and types of arcs of a tangle type. Figure 4 depicts the possible tangle types for loop number (cid:96) = 0 , , (cid:96) = 2 we have three distinct tangle types P L , P C , and P R that all have windingnumber 0.4.3. Saddles.
We now consider saddles S : p ↔ p (cid:48) that relate two planar disk tangles p, p (cid:48) ∈ d m . We definea pair of saddle insertion points ( p , p ) for p and ( p (cid:48) , p (cid:48) ) for p (cid:48) as shown in Figure 5. If both saddle insertionpoints in a pair ( q , q ) lie on the strand component, we define q and q such that q < q . If one saddleinsertion point in a pair ( q , q ) lies on the strand component and the other lies on a circle component, wedefine q to be the point on the strand component and q to be the point on the circle component. For eachsaddle insertion point q , we define a corresponding saddle arc or saddle disk circle s that contains q . Weassign an orientation σ ( s ) ∈ { L, R, U } to each saddle arc s as follows. σ ( s ) = L if τ ( s ) ∈ { e, u } and the saddle attaches to the left side of s , R if τ ( s ) ∈ { e, u } and the saddle attaches to the right side of s , U if τ ( s ) = c .For a saddle disk circle s we define σ ( s ) = U . We define L = R and R = L . We will sometimes denote thetype and orientation of a saddle arc s by using superscripts and subscripts: s τ ( s ) σ ( s ) . For the saddle S : p → p (cid:48) shown in Figure 5, the types and orientations of the saddle arcs are( s ) eR , ( s ) eR , ( s (cid:48) ) eL , ( s (cid:48) ) cU . DAVID BOOZER
Figure 4.
Tangle types for loop number (cid:96) = 0 (left), (cid:96) = 1 (center), and (cid:96) = 2 (right).Earringed arcs are red, unearringed arcs are blue, and circle arcs are black. The subscriptsindicate the winding number for each tangle type. The label in parentheses indicates thetype of each tangle type, as explained in Section 6. Arrows connecting a pair of tangle typesindicate that they are related by saddles. Each arrow is labeled by its corresponding saddletype, as explained in Sections 7 and 8.
Figure 5.
Saddle insertion points and saddle arcs for a saddle S : p → p (cid:48) . Shown are thesaddle insertion points ( p , p ) and saddle arcs ( s , s ) for S in p , and the saddle insertionpoints ( p (cid:48) , p (cid:48) ) and saddle arcs ( s (cid:48) , s (cid:48) ) for S in p (cid:48) .We now consider saddles S : p ↔ p (cid:48) for which the saddle insertion points ( p , p ) lie on distinct saddlearcs ( s , s ), which implies the saddle insertion points ( p (cid:48) , p (cid:48) ) must lie on distinct saddle arcs ( s (cid:48) , s (cid:48) ). Forsuch saddles we enumerate the possibilities for the types and orientations of the saddle arcs, and we indicatethe relationship between the winding number and annular circle number of p and p (cid:48) : Lemma 4.1.
Consider a saddle S : p ↔ p (cid:48) between planar tangles p, p (cid:48) ∈ d m such that the saddle insertionpoints ( p , p ) lie on distinct saddle arcs ( s , s ) of p and the saddle insertion points ( p (cid:48) , p (cid:48) ) lie on distinctsaddle arcs ( s (cid:48) , s (cid:48) ) of p (cid:48) . Without loss of generality we will assume that w ( p (cid:48) ) ≥ w ( p ) and if w ( p ) = w ( p (cid:48) ) HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 9
Figure 6.
Interleaved saddles (
S, T ) and induced saddles S and T . then c a ( p (cid:48) ) ≥ c a ( p ) . The possible such saddles are w ( p (cid:48) ) − w ( p ) c a ( p (cid:48) ) − c a ( p ) ( s , s ) ( s (cid:48) , s (cid:48) )2 0 ( s ) τL < ( s ) τR ( s (cid:48) ) τR < ( s (cid:48) ) τL s ) τσ < ( s ) τσ ( s (cid:48) ) τσ , ( s (cid:48) ) cU s ) cU − ( s ) cU ( s (cid:48) ) cU , ( s (cid:48) ) cU , where τ ∈ { e, u } and σ ∈ { L, R } .Proof. First consider the case where s and s are strand arcs; that is, τ ( s ) = τ ( s ) ∈ { e, u } . Let p A and p (cid:48) A denote the annular 1-tangles corresponding to p and p (cid:48) , and let a denote the portion of the single arcof p A that lies between the saddle insertion points p and p . Let s denote the arc in p connecting p to p that corresponds to the saddle S . Then σ := a ∪ s is a circle in the annulus. If σ is essential, then thesaddle S flips the orientation of a from clockwise in p A to counterclockwise in p (cid:48) A , so w ( p (cid:48) ) − w ( p ) = 2 and c a ( p (cid:48) ) − c a ( p ) = 0. If σ is inessential, then the saddle S splits a circle from p A , hence w ( p (cid:48) ) − w ( p ) = 0 and c a ( p (cid:48) ) − c a ( p ) = 1. The statements regarding the orientations of the saddle arcs are clear, and the statementsregarding the types of the saddle arcs follow from the observation that s must cross the overpass/underpassarc a shown in Figure 1 an odd number of times if σ is essential and an even number of times if σ is inessential.The case where s and s are circle arcs; that is, τ ( s ) = τ ( s ) = c , is clear. (cid:3) Interleaved, nested, and disjoint saddles.
Consider a pair of saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) suchthat the saddle insertion points ( p , p ) for S in p lie on strand arcs ( s , s ) of p , the saddle insertion points( q , q ) for T in p lie on strand arcs ( t , t ) of p , τ ( s ) = τ ( s ) ∈ { e, u } , and τ ( t ) , τ ( t ) ∈ { e, u } . The saddles S and T induce saddles S : ˜ p ↔ p (cid:48)(cid:48) and T : p (cid:48) ↔ p (cid:48)(cid:48) for some planar tangle p (cid:48)(cid:48) , resulting in a commutingsquare of saddles: p p (cid:48) ˜ p p (cid:48)(cid:48) . ST TS
There are three types of such pairs (
S, T ): interleaved, nested, and disjoint.
Definition 4.2.
We say that saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) are interleaved if p < q < p < q , where( p , p ) are the saddle insertion points for S in p and ( q , q ) are the saddle insertion points for T in p , τ ( s ) = τ ( s ) ∈ { e, u } , and τ ( t ) = τ ( t ) ∈ { e, u } .The commuting square for interleaved saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) is depicted in Figure 6. Thefollowing lemma describes the relationships among the planar tangles in this commuting square: Figure 7.
Nested saddles (
S, T ) and induced saddles S and T . Lemma 4.3.
Consider interleaved saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) with induced saddles T : p (cid:48) ↔ p (cid:48)(cid:48) and S : ˜ p ↔ p (cid:48)(cid:48) . The winding numbers and circle numbers of the planar tangles are related by the followingcommutative diagram: p n ± ( r ) p (cid:48) n ( r )˜ p n ( r ) p (cid:48)(cid:48) n ( r + 1) , ST TS where the plus sign in n ± is for σ ( s ) = R and the minus sign is for σ ( s ) = L . The saddles S and T connect the additional circle component of p (cid:48)(cid:48) to opposite sides of the strand (i.e., σ ( s (cid:48)(cid:48) ) = σ ( t (cid:48)(cid:48) ) ∈ { L, R } ).For s = t and s = t we have that τ ( p (cid:48) n ) = τ (˜ p n ) = τ ( p (cid:48)(cid:48) n ) , otherwise τ ( p (cid:48) n ) , τ (˜ p n ) , and τ ( p (cid:48)(cid:48) n ) are alldistinct.Proof. The types and orientations of the saddle arcs are as shown in the following diagram:( s ) τ ( s ) σ ( s ) ≤ ( t ) τ ( t ) σ ( s ) ≤ ( s ) τ ( s ) σ ( s ) ≤ ( t ) τ ( t ) σ ( s ) ( s (cid:48) ) τ ( s ) σ ( s ) ≤ ( t (cid:48) ) τ ( t ) σ ( s ) ≤ ( s (cid:48) ) τ ( s ) σ ( s ) ≤ ( t (cid:48) ) τ ( t ) σ ( s ) (˜ s ) τ ( s ) σ ( s ) ≤ (˜ t ) τ ( t ) σ ( s ) ≤ (˜ s ) τ ( s ) σ ( s ) ≤ (˜ t ) τ ( t ) σ ( s ) ( s (cid:48)(cid:48) ) τ ( s ) σ ( s ) ≤ ( t (cid:48)(cid:48) ) τ ( t ) σ ( s ) , ( s (cid:48)(cid:48) ) cU − ( t (cid:48)(cid:48) ) cU , ST TS where σ ( s ) ∈ { L, R } . The claim regarding the tangle types is clear. (cid:3) Definition 4.4.
We say that saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) are nested if p < q < q < p , where( p , p ) are the saddle insertion points for S in p and ( q , q ) are the saddle insertion points for T in p , τ ( s ) = τ ( s ) ∈ { e, u } , and τ ( t ) = τ ( t ) ∈ { e, u } . Remark 4.5.
It is not possible to have saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) such that q ≤ p < p ≤ q ,where ( p , p ) are the saddle insertion points for S in p , ( q , q ) are the saddle insertion points for T in p , τ ( s ) = τ ( s ) ∈ { e, u } , and τ ( t ) = τ ( t ) ∈ { e, u } .The commuting square for nested saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) is depicted in Figure 7. The followinglemma describes the relationships among the planar tangles in this commuting square: Lemma 4.6.
Consider nested saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) with induced saddles T : p (cid:48) ↔ p (cid:48)(cid:48) and S : ˜ p ↔ p (cid:48)(cid:48) . The winding numbers and circle numbers of the planar tangles are related by the following HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 11
Figure 8.
Disjoint saddles (
S, T ) and induced saddles S and T . commutative diagram: p n ± ( r ) p (cid:48) n ( r )˜ p n ± ( r + 1) p (cid:48)(cid:48) n ( r + 1) , ST TS where the plus sign in n ± is for σ ( s ) = R and the minus sign is for σ ( s ) = L . The saddles T and T connectthe additional circle components of ˜ p and p (cid:48)(cid:48) to opposite sides of the strand (i.e., σ (˜ t ) = σ ( t (cid:48)(cid:48) ) ∈ { L, R } ). For t = t we have that τ ( p n ± ) = τ (˜ p n ± ) and τ ( p (cid:48) n ) = τ ( p (cid:48)(cid:48) n ) , otherwise τ ( p n ± ) (cid:54) = τ (˜ p n ± ) and τ ( p (cid:48) n ) (cid:54) = τ ( p (cid:48)(cid:48) n ) .Proof. The types and orientations of the saddle arcs are as shown in the following diagram:( s ) τ ( s ) σ ( s ) ≤ ( t ) τ ( t ) σ ( t ) < ( t ) τ ( t ) σ ( t ) ≤ ( s ) τ ( s ) σ ( s ) ( s (cid:48) ) τ ( s ) σ ( s ) ≤ ( t (cid:48) ) τ ( t ) σ ( t ) < ( t (cid:48) ) τ ( t ) σ ( t ) ≤ ( s (cid:48) ) τ ( s ) σ ( s ) (˜ s ) τ ( s ) σ ( s ) ≤ (˜ t ) τ ( t ) σ ( t ) ≤ (˜ s ) τ ( s ) σ ( s ) , (˜ t ) cU ( s (cid:48)(cid:48) ) τ ( s ) σ ( s ) < ( t (cid:48)(cid:48) ) τ ( t ) σ ( t ) ≤ ( s (cid:48)(cid:48) ) τ ( s ) σ ( s ) , ( t (cid:48)(cid:48) ) cU . ST TS
The claim regarding the tangle types is clear. (cid:3)
Definition 4.7.
We say that saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) are disjoint if p < p < q < q or q < q < p < p , where ( p , p ) are the saddle insertion points for S in p and ( q , q ) are the saddleinsertion points for T in p , τ ( s ) = τ ( s ) ∈ { e, u } , and τ ( t ) = τ ( t ) ∈ { e, u } .The commuting square for disjoint saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) is depicted in Figure 8. The followinglemma describes the relationships among the planar tangles in this commuting square: Lemma 4.8.
Consider disjoint saddles ( S : p ↔ p (cid:48) , T : p ↔ ˜ p ) with induced saddles T : p (cid:48) ↔ p (cid:48)(cid:48) and S : ˜ p ↔ p (cid:48)(cid:48) . The winding numbers and circle numbers of the planar tangles are related by the followingcommutative diagram: p n ± ( r ) p (cid:48) n ( r )˜ p n ± ( r + 1) p (cid:48)(cid:48) n ( r + 1) , ST TS where the plus sign in n ± is for σ ( s ) = R and the minus sign is for σ ( s ) = L . The saddles T and T connectthe additional circle components of ˜ p and p (cid:48)(cid:48) to the same side of the strand (i.e., σ (˜ t ) = σ ( t (cid:48)(cid:48) ) ∈ { L, R } ). For t = t we have that τ ( p n ± ) = τ (˜ p n ± ) and τ ( p (cid:48) n ) = τ ( p (cid:48)(cid:48) n ) , otherwise τ ( p n ± ) (cid:54) = τ (˜ p n ± ) and τ ( p (cid:48) n ) (cid:54) = τ ( p (cid:48)(cid:48) n ) . Proof.
For the case p < p < q < q , the types and orientations of the saddle arcs are as shown in thefollowing diagram:( s ) τ ( s ) σ ( s ) < ( s ) τ ( s ) σ ( s ) ≤ ( t ) τ ( t ) σ ( t ) < ( t ) τ ( t ) σ ( t ) ( s (cid:48) ) τ ( s ) σ ( s ) < ( s (cid:48) ) τ ( s ) σ ( s ) ≤ ( t (cid:48) ) τ ( t ) σ ( t ) < ( t (cid:48) ) τ ( t ) σ ( t ) (˜ s ) τ ( s ) σ ( s ) < (˜ s ) τ ( s ) σ ( s ) ≤ (˜ t (cid:48) ) τ ( t ) σ ( t ) , (˜ t ) cU ( s (cid:48)(cid:48) ) τ ( s ) σ ( s ) < ( s (cid:48)(cid:48) ) τ ( s ) σ ( s ) ≤ ( t (cid:48)(cid:48) ) τ ( t ) σ ( t ) , ( t (cid:48)(cid:48) ) cU . ST TS
The case q < q < p < p is similar. The claim regarding the tangle types is clear. (cid:3) Chain complex for tangles in the disk
We will now express the chain complex ( C T + , ∂ T + ) in terms of planar tangles in the disk. Given anoriented 1-tangle diagram T in the annulus and an arc a such that ( T, a ) has loop number (cid:96) , let T D denotethe corresponding m -tangle diagram in the disk, where m = (cid:96) + 1, as shown in Figure 1.First we express the bigraded vector space C T + in terms of planar tangles in the disk. Recall that wedefined I = { , } c to be the set of binary strings of length c , where c is the number of crossings of T . Givena binary string i ∈ I , define T i ∈ d m to be the planar tangle obtained by resolving the crossings of T D inthe manner described by i . For each tangle type P ∈ D m , define a bigraded vector space˜ C P = (cid:77) { A ⊗ c ( T i ) [ r ( i ) , r ( i )] | i ∈ I such that τ ( T i ) = P } , (1)where c ( T i ) is the circle number of the planar tangle T i , and a grading-shifted vector space C P by C P = ˜ C P [ h + ( T ) + h + ( P ) , q + ( T ) + q + ( P )] , where h + ( T ) = − n − ( T + ) , q + ( T ) = n + ( T + ) − n − ( T + ) , (2) h + ( P ) = (1 / (cid:96) + w ( P )) , q + ( P ) = (1 / (cid:96) + 3 w ( P )) . (3)The bigraded vector space C T + can then be expressed as C T + = (cid:77) n C n = (cid:77) P C P . Next we express the differential ∂ T + in terms of planar tangles in the disk. We first define several linearmaps by summing over saddles p → p (cid:48) obtained from the resolution of T . For each pair of tangle types P, P (cid:48) ∈ D m , define a linear map T P (cid:48) P : C P → C P (cid:48) by summing the following terms:(1) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) splitting a disk circle or circle arc from a strandarc: ˙ η ⊗ A ⊗ r : A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r + 1 . (2) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) merging a disk circle or circle arc with a strandarc: ˙ (cid:15) ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r. (3) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) splitting a disk circle or circle arc from a diskcircle or circle arc:∆ ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ A ⊗ A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r + 2 . (4) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) merging a disk circle or circle arc with a diskcircle or circle arc: m ⊗ A ⊗ r : A ⊗ A ⊗ A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r + 2 , c ( p (cid:48) ) = r + 1 . For each pair of tangle types
P, P (cid:48) ∈ D m , define a linear map T LP (cid:48) P : C P → C P (cid:48) (respectively T RP (cid:48) P : C P → C P (cid:48) ) by summing the following terms: HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 13 (1) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) splitting a disk circle or circle arc from a strandarc that connects to the strand on the left (respectively right) side:˙ η ⊗ A ⊗ r : A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r + 1 . (2) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) merging a disk circle or circle arc with a strandarc that connects to the strand on the left (respectively right) side:˙ (cid:15) ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r. For each pair of tangle types
P, P (cid:48) ∈ D m , define a linear map ˜ T LP (cid:48) P : C P → C P (cid:48) (respectively ˜ T RP (cid:48) P : C P → C P (cid:48) ) by summing the following terms:(1) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) splitting a disk circle or circle arc from a strandarc that connects to the strand on the left (respectively right) side: η ⊗ A ⊗ r : A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r + 1 . (2) For a saddle p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) merging a disk circle or circle arc with a strandarc that connects to the strand on the left (respectively right) side: (cid:15) ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r. For each pair of tangle types
P, P (cid:48) ∈ D m , define a linear map Q P (cid:48) P : C P → C P (cid:48) by summing the followingterms:(1) If w ( P (cid:48) ) = w ( P ) −
2, then for a saddle p → p (cid:48) :1 A ⊗ r : A ⊗ r → A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r. Define ∂ P = T P P , ˜ ∂ LP = ˜ T LP P , ˜ ∂ RP = ˜ T RP P . Given a saddle S : p → p (cid:48) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) , we define, for example, T P (cid:48) P ( S ) to be the term in T P (cid:48) P corresponding to this saddle, and similarly for the other linear maps. Note that T P (cid:48) P , ˜ T RP (cid:48) P , and ˜ T LP (cid:48) P correspond to saddles p → p (cid:48) that preserve the winding number and change the circle number by one, and Q P (cid:48) P corresponds to saddles that lower the winding number by two and preserve the circle number. Thelinear map ∂ T + : C T + → C T + can be expressed as ∂ T + = (cid:88) n C n = (cid:88) P (cid:88) P (cid:48) T P (cid:48) P = (cid:88) P ∂ P + (cid:88) P (cid:88) P (cid:48) (cid:54) = P T P (cid:48) P , (4)and the linear map ∂ + T + : C T + → C T + can be expressed as ∂ + T + = (cid:88) n ( ˜ ∂ Ln − Q n + Q n ˜ ∂ Ln ) = (cid:88) P (cid:88) P (cid:48) (cid:88) P (cid:48)(cid:48) ( ˜ T LP (cid:48)(cid:48) P (cid:48) Q P (cid:48) P + Q P (cid:48)(cid:48) P (cid:48) ˜ T LP (cid:48) P ) . (5)Recall that the differential ∂ T + : C T + → C T + is given by ∂ T + = ∂ T + + ∂ + T + .Having reformulated the chain complex ( C T + , ∂ T + ) in terms of planar tangles in the disk, we can nowreturn to the task of completing the proof of Theorem 3.1. Recall that we started with an oriented annular1-tangle diagram T with loop number (cid:96) . We flipped the outermost loop of T under the annulus to obtain anannular 1-tangle diagram T with loop number (cid:96) − C T + , ∂ T + ) for the link diagram T + . Our goal is to use the Reduction Lemma3.2 to show that ( C T + , ∂ T + ) is chain-homotopy equivalent to ( C T + , ∂ T + ). The first step is to understand thestructure of ( C T + , ∂ T + ) as expressed in terms of planar tangles in the disk.We first consider the vector space C T + . For each tangle type P ∈ D m , where m = (cid:96) + 1, we definetangle types [ P ] , [ P ] ∈ D m − as shown in Figure 9 by taking the 0-resolution and 1-resolution of the oneadditional crossing in T . We refer to [ P ] and [ P ] as the and of P . We refer tothe arcs r and r shown in Figure 9 as resolution arcs . We can then express C T + as C T + = (cid:77) P C P , (6) Figure 9.
The 0-resolution [ P ] and 1-resolution [ P ] of a tangle type P .where C P = C [ P ] ⊕ C [ P ] . Next we consider the differential ∂ T + . We first want to identify the various linear maps corresponding tothe saddles obtained from the resolution of T . We classify these saddles into three types:(1) Saddles [ p ] → [ p ] obtained by resolving the one additional crossing of T .(2) Saddles [ p ] → [ p (cid:48) ] and [ p ] → [ p (cid:48) ] that are induced by a type-preserving (i.e. τ ( p ) = τ ( p (cid:48) )) saddle S : p → p (cid:48) obtained from the resolution of T .(3) Saddles [ p ] → [ p (cid:48) ] and [ p ] → [ p (cid:48) ] that are induced by a type-changing (i.e. τ ( p ) (cid:54) = τ ( p (cid:48) )) saddle S : p → p (cid:48) obtained from the resolution of T .For each tangle type P , we define a linear map ∂ P : C P → C P by keeping only those terms of ∂ T + that map C P to C P ; such terms are maps or products of pairs of maps corresponding to saddles of types (1) and (2).The map ∂ P has the form ∂ P = (cid:18) ∂ [ P ] ∂ [ P ] [ P ] ∂ [ P ] (cid:19) , where ∂ [ P ] [ P ] : C [ P ] → C [ P ] consists of those terms of ∂ P that map C [ P ] to C [ P ] . In Section 6, wedescribe ( C P , ∂ P ) for each tangle type P ∈ D m . Remark 5.1.
We show in Lemma 11.1 below that ( ∂ T + ) = 0, but it is generally not the case that ( ∂ P ) = 0,so ( C P , ∂ P ) is not a chain complex.The remaining saddles obtained from the resolution of T are of type (3), and are induced by saddles S : p → p (cid:48) obtained from the resolution of T that change the tangle type (i.e., τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) (cid:54) = P ).For each type-changing saddle S : p → p (cid:48) we define a vector space C S = C P ⊕ C P (cid:48) . We define a linear map ∂ S : C S → C S by keeping only those terms of ∂ T + that are maps or products ofmaps corresponding to saddles used to define ∂ P or ∂ P (cid:48) , or to saddles [ p ] → [ p (cid:48) ] or [ p ] → [ p (cid:48) ] induced by S : p → p (cid:48) . We describe ( C S , ∂ S ) for each type-changing saddle S : p → p (cid:48) . In Section 7 we describe saddlesthat change the winding number by two and preserve the circle number, and in Section 8 we describe saddlesthat preserve the winding number and change the annular circle number by one; according to Lemma 4.1,these are the only two possibilities. In Sections 9 and 10 we use the results of Sections 6, 7, and 8 to reducethe chain complex ( C T + , ∂ T + ) and thereby complete our proof of Theorem 3.1.We end this section with a few observations regarding the linear maps ∂ T + and ∂ + T + defined in equations(4) and (5) that will be useful in what follows: Remark 5.2.
The linear map ∂ T + is a sum of maps corresponding to saddles obtained in the resolution of T that preserve the winding number. Remark 5.3.
The linear map ∂ + T + is a sum of products of pairs of maps corresponding to pairs of saddlesobtained from the resolution of T . The pairs of saddles whose corresponding maps are included in the sum HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 15 comprise two adjacent sides of a commuting square of saddles corresponding to interleaved saddles (
S, T ) ofthe following forms: p n ( r ) p (cid:48) n − ( r )˜ p n − ( r ) p (cid:48)(cid:48) n − ( r + 1) , ST TS p (cid:48)(cid:48) n − ( r ) p (cid:48) n ( r )˜ p n ( r ) p n ( r + 1) , ST TS (7)and nested and disjoint saddles (
S, T ) of the following forms: p n ( r ) p (cid:48) n − ( r )˜ p n ( r + 1) p (cid:48)(cid:48) n − ( r + 1) , ST TS p (cid:48)(cid:48) n − ( r ) p (cid:48) n ( r )˜ p n − ( r + 1) p n ( r + 1) , ST TS (8) p (cid:48) n ( r ) p (cid:48)(cid:48) n − ( r ) p n ( r + 1) ˜ p n − ( r + 1) , ST TS p (cid:48) n − ( r ) p n ( r ) p (cid:48)(cid:48) n − ( r + 1) ˜ p n ( r + 1) . ST TS (9)We can thus express ∂ + T + in terms of a sum over all interleaved, nested, and disjoint pairs of saddles ( S, T )obtained from the resolution of T . In fact, as we show in Corollary 5.5 below, disjoint saddles do notcontribute, so it suffices to sum over interleaved and nested saddles.As an application of Remark 5.3, we prove the following lemma, which shows we could just as well useright-handed maps, as opposed to left-handed maps, in the definition of ∂ + T + : Lemma 5.4.
We have (cid:88) P ( ˜ T LP (cid:48)(cid:48) P Q P P + Q P (cid:48)(cid:48) P ˜ T LP P ) = (cid:88) P ( ˜ T RP (cid:48)(cid:48) P Q P P + Q P (cid:48)(cid:48) P ˜ T RP P ) . (10) Proof.
The left and right sides of equation (10) are given by summing products of maps in the commutingsquare corresponding to interleaved saddles (
S, T ) of the forms shown in diagram (7) and nested and disjointsaddles (
S, T ) of the forms shown in diagrams (8) and (9), where τ ( p ) = P and τ ( p (cid:48)(cid:48) ) = P (cid:48)(cid:48) . We claim thatin each case the contribution of the pair ( S, T ) to the left and right sides of equation (10) is the same. Wewill show this for a few representative examples.For the first form of interleaved saddles (
S, T ) shown in diagram (7), the contributions of the correspondingcommuting square of saddles to the left and right sides of equation (10) are˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + ˜ T LP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) , ˜ T RP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + ˜ T RP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) . From Lemma 4.3 for interleaved saddles, it follows that˜ T LP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) = ˜ T RP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) , ˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = ˜ T RP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) = 0 . So the contribution of (
S, T ) to the left and right sides of equation (10) is the same.For the first form of nested saddles (
S, T ) shown in diagram (8), the contributions of the correspondingcommuting square of saddles to the left and right sides of equation (10) are˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T L ˜ P P ( T ) , ˜ T RP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T R ˜ P P ( T ) . From Lemma 4.6 for nested saddles, it follows that there is an orientation σ ∈ { L, R } such that˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T σ ˜ P P ( T ) , Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T σ ˜ P P ( T ) = ˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = 0 . So the contribution of (
S, T ) to the left and right sides of equation (10) is the same.For the first form of disjoint saddles (
S, T ) shown in diagram (8), the contributions of the correspondingcommuting square of saddles to the left and right sides of equation (10) are˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T L ˜ P P ( T ) , ˜ T RP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T R ˜ P P ( T ) . From Lemma 4.8 for disjoint saddles, it follows that there is an orientation σ ∈ { L, R } such that˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T σ ˜ P P ( T ) , Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T σ ˜ P P ( T ) = ˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = 0 . So the contribution of (
S, T ) to the left and right sides of equation (10) is zero. (cid:3)
The proof of Lemma 5.4 also proves:
Corollary 5.5.
Pairs of disjoint saddles do not contribute to ∂ + T + . Reduction of planar tangles
We classify each planar tangle p and tangle type P according to the types and relative positions of thearcs B and C : Type P1A : τ ( B ) = τ ( C ) ∈ { e, u } and B < C,
Type P1B : τ ( B ) = τ ( C ) ∈ { e, u } and C < B,
Type P2 : τ ( B ) = τ ( C ) = c. We say a planar tangle is type P1 if it is either type P1A or type P1B. The following lemma describes therelationship between a planar tangle p and its 0-resolution and 1-resolution [ p ] and [ p ] : Lemma 6.1.
For a planar tangle p with winding number w ( p ) = n and circle number c ( p ) = r , the windingnumber and circle number of the planar tangles [ p ] and [ p ] are given by p w ([ p ] ) w ([ p ] ) c ([ p ] ) c ([ p ] )P1A n + 1 n + 1 r + 1 r P1B n − n − r r + 1P2 n + 1 n − r − r − Proof.
First consider a type P1A planar tangle p . The union of the reduction arc r and the segment ofthe strand starting at point m + 2 on the disk and ending at the point m + 1 on the disk constitutes a newcircle component of [ p ] , hence c ([ p ] ) = c ( p ) + 1. The saddle [ p ] → [ p ] connects this circle componentto the strand component, hence c ([ p ] ) = c ( p ) and w ([ p ] ) = w ([ p ] ). Arc C is oriented inwards at point m + 1, so in going from p to [ p ] and [ p ] we are removing from p a single loop that winds clockwise, hence w ([ p ] ) = w ([ p ] ) = w ( p ) + 1. The case of a type P1B planar tangle is similar.Next consider a type P2 planar tangle p . Since the reduction arc r connects the circle component of p containing B and C to the strand, we have that c a ([ p ] ) = c a ( p ) −
1. The reduction arc r connects points m + 1 and m + 2 on the disk, and the point m + 1 on the disk is oriented outwards, so the addition ofthe reduction arc r to p extends the strand component by adding loops with net winding number 1. Thus w ([ p ] ) = w ( p ) + 1. Similarly c a ([ p ] ) = c a ( p ) − w ([ p ] ) = w ( p ) − (cid:3) We can use Lemma 6.1 to relate the vector spaces C P , C [ P ] , and C [ P ] for the tangle types P , [ P ] , and[ P ] . We will initially view these as ungraded vector spaces, then in Lemma 6.7 we consider the bigradings.If P is of type P1A, then [ P ] has one more circle component than P and [ P ] , so as ungraded vector spaces C [ P ] = C P ⊗ A, C [ P ] = C P . If P is of type P1B, then [ P ] has one more circle component than P and [ P ] , so as ungraded vector spaces C [ P ] = C P , C [ P ] = C P ⊗ A. If P is of type P2, then P has one more circle component than [ P ] and [ P ] , so can define a vector space W such that as ungraded vector spaces C P = W ⊗ A and C [ P ] = W, C [ P ] = W. We can also relate the linear maps ∂ P , ∂ [ P ] , and ∂ [ P ] for the tangle types P , [ P ] , and [ P ] . Recall fromLemma 6.1 that if P is of type P1A then [ P ] has one more circle component than P and [ P ] , if P is oftype P1B then [ P ] has one more circle component than P and [ P ] , and if P is of type P2 then P has onemore circle component than [ P ] and [ P ] . Given a tangle type P of type P1A, P1B, or P2, define linearmaps ˜ ∂ c P : C P → C P , ˜ ∂ c P : C P → C P , or ˜ ∂ cP : W → W by summing the following terms over saddles p → p (cid:48) obtained from the resolution of T : HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 17 (1) For a saddle p → p (cid:48) with τ ( p ) = τ ( p (cid:48) ) = P splitting a disk circle from a circle arc in the additionalcircle component of [ P ] , [ P ] , or P : η ⊗ A ⊗ r : A ⊗ r → A ⊗ A ⊗ r , c ( p ) = r, c ( p (cid:48) ) = r + 1 . (2) For a saddle p → p (cid:48) with τ ( p ) = τ ( p (cid:48) ) = P merging a disk circle and a circle arc in the additionalcircle component of [ P ] , [ P ] , or P : (cid:15) ⊗ A ⊗ r : A ⊗ A ⊗ r → A ⊗ r , c ( p ) = r + 1 , c ( p (cid:48) ) = r. For a tangle type P of type P1, we can use the linear maps ˜ ∂ c P and ˜ ∂ c P , together with the fact that m = ˙ (cid:15) ⊗ A + (cid:15) ⊗ x and ∆ = ˙ η ⊗ A + η ⊗ x , to express the linear maps ∂ [ P ] and ∂ [ P ] in terms of ∂ P . If P is of type P1A, then( C [ P ] , ∂ [ P ] ) = ( C P ⊗ A, ∂ P ⊗ A + ˜ ∂ c P ⊗ x ) , ( C [ P ] , ∂ [ P ] ) = ( C P , ∂ P ) . If P is of type P1B, then( C [ P ] , ∂ [ P ] ) = ( C P , ∂ P ) , ( C [ P ] , ∂ [ P ] ) = ( C P ⊗ A, ∂ P ⊗ A + ˜ ∂ c P ⊗ x ) . The relationship between ∂ P , ∂ [ P ] , and ∂ [ P ] for a tangle type P of type P2 is described by Lemma 6.3below.Recall from Section 5 that for each tangle type P we defined a vector space C P and a linear map ∂ P : C P → C P by C P = C [ P ] ⊕ C [ P ] , ∂ P = (cid:18) ∂ [ P ] ∂ [ P ] [ P ] ∂ [ P ] (cid:19) , where ∂ [ P ] [ P ] : C [ P ] → C [ P ] consists of those terms of ∂ P that map C [ P ] to C [ P ] . The relationshipbetween ( C P , ∂ P ) and ( C P , ∂ P ) is described by the following results: Lemma 6.2.
For a type P1A tangle type P , we have C P = ( C P ⊗ A ) ⊕ C P , ∂ P = (cid:18) ∂ P ⊗ A + ˜ ∂ c P ⊗ x C P ⊗ ˙ (cid:15) ∂ P (cid:19) , where ˜ ∂ c P : C P → C P is the linear map corresponding to saddles connecting disk circles to arcs in P that lieon the additional circle component of [ P ] . For a type P1B tangle type P , we have C P = C P ⊕ ( C P ⊗ A ) , ∂ P = (cid:18) ∂ P C P ⊗ ˙ η ∂ P ⊗ A + ˜ ∂ c P ⊗ x (cid:19) , where ˜ ∂ c P : C P → C P is the linear map corresponding to saddles connecting disk circles to arcs in P that lieon the additional circle component of [ P ] .Proof. We will prove the claim for P of type P1A; the case of type P1B is similar. We have that C P = C [ P ] ⊕ C [ P ] = ( C P ⊗ A ) ⊕ C P . The linear maps C [ P ] → C [ P ] corresponding to saddles [ p ] → [ p ] obtained by resolving the one additionalcrossing of T are described by the diagram C [ P ] = ( C P ⊗ A ) n +1 C [ P ] = ( C P ) n +1 , T [ P ]1[ P ]0 =1 CP ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P ]0 =1 CP ⊗ (cid:15) where n = w ( P ) is the winding number of P . Note that such saddles merge the additional circle componentof [ p ] with the strand. The claim now follows from the definition of ∂ T + . (cid:3) Lemma 6.3.
For each type P2 tangle type P , we have ( C P , ∂ P ) = ( C P , ∂ P ) . Proof.
We define (
W, ∂ W ) := ( C [ P ] , ∂ [ P ] ) = ( C [ P ] , ∂ [ P ] ) . Then C P = C [ P ] ⊕ C [ P ] = ( W ) n +1 ⊕ ( W ) n − , where n = w ( P ) is the winding number of P . The linear map C [ P ] → C [ P ] corresponding to saddles[ p ] → [ p ] obtained by resolving the one additional crossing of T is described by the diagram C [ P ] = ( W ) n +1 C [ P ] = ( W ) n − . Q [ P ]1[ P ]0 =1 W From the definition of ∂ T + , it follows that ∂ P = (cid:18) ∂ W ∂ L [ P ] Q [ P ] [ P ] + Q [ P ] [ P ] ˜ ∂ L [ P ] ∂ W (cid:19) . Substituting Q [ P ] [ P ] = 1 W , we obtain˜ ∂ L [ P ] Q [ P ] [ P ] + Q [ P ] [ P ] ˜ ∂ L [ P ] = ˜ ∂ L [ P ] + ˜ ∂ L [ P ] . We claim that ˜ ∂ L [ P ] + ˜ ∂ L [ P ] = ˜ ∂ cW , (11)where ˜ ∂ cW : W → W is the linear map corresponding to saddles connecting disk circles to circle arcs inplanar tangles p of type τ ( p ) = P that lie on the additional circle component of p that is not present in [ p ] or [ p ] . We can see this as follows. Consider a saddle connecting a disk circle to an arc a of a planar tangle p with tangle type τ ( p ) = P . The arc a of p lies on a unique arc [ a ] of [ p ] and [ a ] of [ p ] . If a lies in thestrand component of p we have σ ( a ) = σ ([ a ] ) = σ ([ a ] ) ∈ { L, R } , so such saddles do not contribute to either side of equation (11). If a lies in the additional annular circle of p , which contains the arcs B and C , we have σ ( a ) = U, σ ([ a ] ) = σ ([ a ] ) ∈ { L, R } , so such saddles contribute equally to the left and right sides of equation (11). If a lies in an annular circleof p that does not contain the arcs B and C , and hence is also present in [ p ] and [ p ] , we have σ ( a ) = σ ([ a ] ) = σ ([ a ] ) = U, so such saddles do not contribute to either side of equation (11). So in each case the contribution to bothsides of equation (11) is the same.We can identify W ⊗ A := ( W ) n +1 and W ⊗ x := ( W ) n − to obtain C P = ( W ) n +1 ⊕ ( W ) n − = W ⊗ A = C P , ∂ P = (cid:18) ∂ W ∂ cW ∂ W (cid:19) = ∂ W ⊗ A + ˜ ∂ cW ⊗ x = ∂ P . (cid:3) Recall from equation (6) that the vector space C T + for the chain complex ( C T + , ∂ T + ) is the direct sum ofthe spaces C P over all tangle types P . The differential ∂ T + contains all of the terms of ∂ P for each tangletype P . Lemma 6.2 shows that the chain complex ( C T + , ∂ T + ) is such that for each tangle type P of typetype P1 we can apply the Reduction Lemma 3.2. In particular, for P of type P1A we can reduce on the map1 C P ⊗ ˙ (cid:15) : C P ⊗ A → C P in ∂ P , and for P of type P1B we can reduce on the map 1 C P ⊗ ˙ η : C P → C P ⊗ x in ∂ P . We will refer to such a reduction as a reduction on P . We can thus make the following definition: Definition 6.4.
We define ( C red , ∂ red ) to be the chain complex that results from reducing ( C T + , ∂ T + ) on P for each type P1 planar tangle P (i.e, reducing on the map 1 C P ⊗ ˙ (cid:15) : C P ⊗ A → C P for P of type P1Aand reducing on the map 1 C P ⊗ ˙ η : C P → C P ⊗ x for P of type P1B). HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 19
For a type P1A tangle type P , we identify C P with the subspace C P ⊗ x of C P = ( C P ⊗ A ) ⊕ C P . Definea subspace V P of C T + by V P = (cid:77) P (cid:48) (cid:54) = P C P (cid:48) ⊕ ( C P ⊗ x ) . Linear maps α : V P → C P ⊗ A , β : C P → V P that occur in ∂ T + are eliminated when we reduce on P , and linear maps α : V P → C P , β : C P ⊗ A → V P that occur in ∂ T + yield reduction factors when we reduce on P .For a type P1B tangle type P , we identify C P with the subspace C P ⊗ A of C P = C P ⊕ ( C P ⊗ A ). Definea subspace V P of C T + by V P = (cid:77) P (cid:48) (cid:54) = P C P (cid:48) ⊕ ( C P ⊗ A ) . Linear maps α : V P → C P , β : C P ⊗ x → V P that occur in ∂ T + are eliminated when we reduce on P , and linear maps α : V P → C P ⊗ x, β : C P → V P that occur in ∂ T + yield reduction factors when we reduce on P .Thus, from Lemmas 6.2 and 6.3 we obtain the following lemma: Lemma 6.5.
For each type P1A tangle type P , we obtain a residual term ∂ P and a reduction factor β ( P ) = ˜ ∂ c P : C P → C P . For each type P1B tangle type P , we obtain a residual term ∂ P and reduction factor α ( P ) = ˜ ∂ c P : C P → C P . For each type P2 tangle P , we obtain a residual term ∂ P and no reduction factors. To complete the proof of Theorem 3.1, we want to show that ( C red , ∂ red ) = ( C T + , ∂ T + ). Lemmas 6.2 and6.3 show that Lemma 6.6.
We have C red = C T + as ungraded vector spaces. We now discuss the bigrading of the vector space C P . Recall that in equation (1) we defined a bigradedvector space ˜ C P . The following lemma explains the grading shifts described in equations (2) and (3): Lemma 6.7.
For a tangle type P of loop number (cid:96) , the bigraded vector space C P is given by C P = ˜ C P [ h + ( T, P ) , q + ( T, P )] , where h + ( T, P ) = h + ( T ) + h + ( P ) , h + ( T ) = − n − ( T + ) , h + ( P ) = (1 / (cid:96) + w ( P )) , (12) q + ( T, P ) = q + ( T ) + q + ( P ) , q + ( T ) = n + ( T + ) − n − ( T + ) , q + ( P ) = (1 / (cid:96) + 3 w ( P )) . (13) Proof.
We will prove the claim by induction on the loop number (cid:96) . It is clearly true for the base case (cid:96) = 0,since there is a unique tangle type P , as shown in Figure 4, and C P = ˜ C P [ h + ( T, P ) , q + ( T, P )] = ˜ C P [ h + ( T ) , q + ( T )]is the underlying bigraded vector space of the chain complex for the reduced Khovanov homology of the link T + = T − .For the induction step, assume the claim is true for loop number (cid:96) −
1, and let P be a tangle type withloop number (cid:96) . By the induction hypothesis, the bigrading of C P is given by C P = C [ P ] ⊕ C [ P ] [1 ,
1] = ˜ C [ P ] [ h + ( T, [ P ] ) , q + ( T, [ P ] )] ⊕ ˜ C [ P ] [ h + ( T, [ P ] ) + 1 , q + ( T, [ P ] ) + 1] . (14) We will use equation (14) to determine the bigrading of C P for each tangle type P .If P is of type P1A, then˜ C [ P ] = ˜ C P ⊗ A, ˜ C [ P ] = ˜ C P . When we reduce on P , we identify C P as a grading-shifted version of the subspace ˜ C P ⊗ x of ˜ C [ P ] , and x has quantum grading −
1, so from equation (14) it follows that C P = ˜ C P [ h + ( T, [ P ] ) , q + ( T, [ P ] ) − . The loop number of [ P ] is (cid:96) −
1, and by Lemma 6.1 the winding number of [ P ] is w ([ P ] ) = w ( P ) + 1, soequations (12) and (13) imply that( h + ( T, [ P ] ) , q + ( T, [ P ] ) −
1) = ( h + ( T, P ) , q + ( T, P )) . (15)Thus the claim holds if P is of type P1A.If P is of type P1B, then˜ C [ P ] = ˜ C P , ˜ C [ P ] = ˜ C P ⊗ A. When we reduce on P , we identify C P as a grading-shifted version of the subspace ˜ C P ⊗ A of ˜ C [ P ] , and1 A has quantum grading 1, so from equation (14) it follows that C P = ˜ C P [ h + ( T, [ P ] ) + 1 , q + ( T, [ P ] ) + 2] . The loop number of [ P ] is (cid:96) −
1, and by Lemma 6.1 the winding number of [ P ] is w ([ P ] ) = w ( P ) −
1, soequations (12) and (13) imply that( h + ( T, [ P ] ) + 1 , q + ( T, [ P ] ) + 2) = ( h + ( T, P ) , q + ( T, P )) . (16)Thus the claim holds if P is of type P1B.If P is of type P2, then we can define a bigraded vector space W such that˜ C P = W ⊗ A = ( W ⊗ A ) ⊕ ( W ⊗ x ) , ˜ C [ P ] = W, ˜ C [ P ] = W. (17)The loop numbers of [ P ] and [ P ] are (cid:96) −
1, and by Lemma 6.1 the winding numbers of [ P ] and [ P ] are w ([ P ] ) = w ( P ) + 1 and w ([ P ] ) = w ( P ) −
1, so equations (15) and (16) hold for [ P ] and [ P ] . We have C P = C P as bigraded vector spaces, so from equations (14), (15), (16), and (17), it follows that C P = C P = W [ h + ( T, P ) , q + ( T, P ) + 1] ⊕ W [ h + ( T, P ) , q + ( T, P ) −
1] = ˜ C P [ h + ( T, P ) , q + ( T, P )] , where we have used the fact that the quantum gradings of 1 A and x are 1 and −
1. Thus the claim holds if P is of type P2. (cid:3) Thus we obtain:
Lemma 6.8.
We have C red = C T + as bigraded vector spaces. Reduction of type-changing saddles p n +1 ( r ) ↔ p (cid:48) n − ( r )Recall from Section 5 that for each type-changing saddle S : p → p (cid:48) obtained from the resolution of T wedefined a vector space C S and a linear map ∂ S : C S → C S . The vector space C S is given by C S = C P ⊕ C P (cid:48) for P = τ ( p ) and P (cid:48) = τ ( p (cid:48) ). The linear map ∂ S is obtained by keeping only those terms of ∂ T + that arelinear maps or products of linear maps corresponding to saddles used to define ∂ P or ∂ P (cid:48) , or to saddles[ p ] → [ p (cid:48) ] or [ p ] → [ p (cid:48) ] induced by S : p → p (cid:48) .For a type-changing saddle p n +1 ( r ) ↔ p (cid:48) n − ( r ), Lemma 4.1 implies that the saddle arcs ( s , s ) in p n +1 ( r )are such that τ ( s ) = τ ( s ) ∈ { e, u } . We classify saddles p n +1 ( r ) ↔ p (cid:48) n − ( r ) into the following types, HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 21 depending on the types and relative positions of the arcs s , s , B , and C of p n +1 ( r ):Type W1: τ ( B ) = τ ( C ) ∈ { e, u } and either B < C ≤ s < s or C < B ≤ s < s ,Type W2: τ ( B ) = τ ( C ) ∈ { e, u } and either s ≤ B < C ≤ s or s ≤ C < B ≤ s ,Type W3: τ ( B ) = τ ( C ) ∈ { e, u } and either s < s ≤ B < C or s < s ≤ C < B ,Type W4: τ ( B ) = τ ( C ) = c .For a saddle S : p → p (cid:48) , the linear map ∂ S has the form ∂ S = (cid:18) ∂ P ∂ P (cid:48) P ( S ) ∂ P (cid:48) (cid:19) for some linear map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) . For each type of saddle, we describe the residual terms andreduction factors obtained from ∂ P (cid:48) P ( S ). (Recall that in Lemma 6.5 we described the residual terms andreduction factors obtained from ∂ P and ∂ P (cid:48) .) Lemma 7.1.
Type W1 saddles p n +1 ( r ) ↔ p (cid:48) n − ( r ) do not actually occur.Proof. We will prove the claim for the case
B < C ; the case
C < B is similar. We observe that the union ofthe resolution arc r and the segment of the strand starting at point m + 2 on arc C and ending at point m + 1 on arc D is an inessential circle in the annulus, so a saddle connecting two arcs in this segment cannotchange the winding number. (cid:3) Lemma 7.2.
Type W2 saddles S : p n +1 ( r ) → p (cid:48) n − ( r ) must have C < B .Proof.
For a type W2 saddle the arcs B and C lie between the saddle arcs s and s , hence the order ofthe arcs B and C in P must be opposite to the order of the arcs B (cid:48) and C (cid:48) in P (cid:48) . The saddle p n +1 ( r ) ↔ p (cid:48) n − ( r ) induces saddles [ p n +1 ( r )] k ↔ [ p (cid:48) n − ( r )] k for k = 0 ,
1. By Lemma 6.1, if
C < B and B (cid:48) < C (cid:48) then w ([ p (cid:48) n − ( r )] k ) − w ([ p n +1 ( r )] k ) = 0, whereas if B < C and C (cid:48) < B (cid:48) then w ([ p (cid:48) n − ( r )] k ) − w ([ p n +1 ( r )] k ) = − p n +1 ( r )] k ↔ [ p (cid:48) n − ( r )] k connect circles components to the strand. (cid:3) Lemma 7.3.
For a type W2 saddle S : p n +1 ( r ) → p (cid:48) n − ( r ) we have the following terms. There are noresidual terms. Reduction on P n +1 yields the reduction factor β ( P, S ) = Q P (cid:48) P ( S ) : C P → C P (cid:48) . Reduction on P (cid:48) n − yields the reduction factor α ( P (cid:48) , S ) = Q P (cid:48) P ( S ) : C P → C P (cid:48) . For a type W2 saddle S : p (cid:48) n − ( r ) → p n +1 ( r ) , there are no reduction factors or residual terms.Proof. We have C S = C P ⊕ C P (cid:48) , C P = ( C P ) n ⊕ ( C P ⊗ A ) n , C P (cid:48) = ( C P (cid:48) ⊗ A ) n ⊕ ( C P (cid:48) ) n . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles[ p n +1 ( r )] ↔ [ p (cid:48) n − ( r )] and [ p n +1 ( r )] ↔ [ p (cid:48) n − ( r )] induced by S : p n +1 ( r ) ↔ p (cid:48) n − ( r ): C [ P ] = ( C P ) n C [ P (cid:48) ] = ( C P (cid:48) ⊗ A ) n C [ P ] = ( C P ⊗ A ) n C [ P (cid:48) ] = ( C P (cid:48) ) n . T [ P ]1[ P ]0 =1 CP ⊗ ˙ η T [ P (cid:48) ]0[ P ]0 = Q P (cid:48) P ( S ) ⊗ ˙ η ˜ T L [ P (cid:48) ]0[ P ]0 = Q P (cid:48) P ( S ) ⊗ η T [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ ˙ (cid:15) ˜ T L [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ (cid:15)T [ P ]0[ P (cid:48) ]0 = P PP (cid:48) ( S ) ⊗ ˙ (cid:15) ˜ T L [ P ]0[ P (cid:48) ]0 = P PP (cid:48) ( S ) ⊗ (cid:15)T [ P (cid:48) ]1[ P ]1 = Q P (cid:48) P ( S ) ⊗ ˙ (cid:15)T [ P ]1[ P (cid:48) ]1 = P PP (cid:48) ( S ) ⊗ ˙ η (18) For a saddle S : p n +1 ( r ) → p (cid:48) n − ( r ), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is given by ∂ P (cid:48) P ( S ) = (cid:18) Q P (cid:48) P ( S ) ⊗ ˙ η Q P (cid:48) P ( S ) ⊗ ˙ (cid:15) (cid:19) . The claim for a saddle S : p n +1 ( r ) → p (cid:48) n − ( r ) now follows from reading off terms in ∂ P (cid:48) P ( S ). For a saddle S : p (cid:48) n − ( r ) → p n +1 ( r ), the map ∂ P P (cid:48) ( S ) : C P (cid:48) → C P is given by ∂ P P (cid:48) ( S ) = (cid:18) P P P (cid:48) ( S ) ⊗ ˙ (cid:15) P P P (cid:48) ( S ) ⊗ ˙ η (cid:19) . The claim for a saddle S : p (cid:48) n − ( r ) → p n +1 ( r ) now follows from reading off terms in ∂ P P (cid:48) ( S ). Note, forexample, that the map α ( P, S ) := P P P (cid:48) ( S ) ⊗ ˙ (cid:15) : C P (cid:48) ⊗ A → C P is eliminated when we reduce on P , andthe map β ( P (cid:48) , S ) : P P P (cid:48) ( S ) ⊗ ˙ η : C P (cid:48) → C P ⊗ x is eliminated when we reduce on P (cid:48) . (cid:3) Lemma 7.4.
For a type W3 or W4 saddle S : p n +1 ( r ) → p (cid:48) n − ( r ) , we have the following terms. There areno reduction factors. We have the residual term ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP : C P → C P (cid:48) . For a type W3 or W4 saddle S : p (cid:48) n − ( r ) → p n +1 ( r ) , there are no reduction factors or residual terms.Proof. First consider a type W3 saddle. We will prove the claim for the case
B < C ; the case
C < B issimilar. We have C S = C P ⊕ C P (cid:48) , C P = ( C P ⊗ A ) n +2 ⊕ ( C P ) n +2 , C P (cid:48) = ( C P (cid:48) ⊗ A ) n ⊕ ( C P (cid:48) ) n . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles[ p n +1 ( r )] ↔ [ p (cid:48) n − ( r )] and [ p n +1 ( r )] ↔ [ p (cid:48) n − ( r )] induced by S : p n +1 ( r ) ↔ p (cid:48) n − ( r ): C [ P ] = ( C P ⊗ A ) n +2 C [ P (cid:48) ] = ( C P (cid:48) ⊗ A ) n C [ P ] = ( C P ) n +2 C [ P (cid:48) ] = ( C P (cid:48) ) n . T [ P (cid:48) ]0[ P ]0 =1 CP ⊗ ˙ (cid:15) ˜ T L [ P (cid:48) ]0[ P ]0 =1 CP ⊗ (cid:15) Q [ P (cid:48) ]0[ P ]0 = Q P (cid:48) P ( S ) ⊗ A T [ P (cid:48) ]1[ P ]1 =1 CP (cid:48) ⊗ ˙ (cid:15) ˜ T L [ P (cid:48) ]1[ P ]1 =1 CP (cid:48) ⊗ (cid:15)Q [ P (cid:48) ]1[ P ]1 = Q P (cid:48) P ( S ) (19)For a type W3 saddle S : p n +1 ( r ) → p (cid:48) n − ( r ), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is ∂ P (cid:48) P ( S ) = (cid:32) ˜ ∂ L [ P (cid:48) ] Q [ P (cid:48) ] [ P ] + Q [ P (cid:48) ] [ P ] ˜ ∂ L [ P ]
00 ˜ ∂ L [ P (cid:48) ] Q [ P (cid:48) ] [ P ] + Q [ P (cid:48) ] [ P ] ˜ ∂ L [ P ] (cid:33) . The fact that the matrix element ( C P ⊗ A ) n +2 → ( C P (cid:48) ) n is zero follows from the fact that the saddles( S : [ p n +1 ( r )] → [ p (cid:48) n − ( r )] , T : [ p n +1 ( r )] → [ p n +1 ( r )] ) are disjoint.We claim that ˜ ∂ L [ P (cid:48) ] Q [ P (cid:48) ] [ P ] + Q [ P (cid:48) ] [ P ] ˜ ∂ L [ P ] = ( ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP ) ⊗ A . (20)We can see this as follows. Let ( p , p ) denote the saddle insertion points of S in p n +1 . Consider a saddleconnecting a disk circle to an arc a of p n +1 with saddle insertion point q ∈ a . There is a correspondingsaddle connecting the disk circle to an arc a (cid:48) of p (cid:48) n − with saddle insertion point q (cid:48) ∈ a (cid:48) . The arc a of p n +1 lies on a unique arc or disk circle [ a ] of [ p n +1 ] . The arc a (cid:48) of p (cid:48) n − lies on a unique arc or disk circle [ a (cid:48) ] of [ p (cid:48) n − ] . If the arc a is a strand arc, the orientations of these arcs are as follows: q < p : σ ( a ) = σ ([ a ] ) = σ ( a (cid:48) ) = σ ([ a (cid:48) ] ) ∈ { L, R } ,p < q < p : σ ( a ) = σ ([ a ] ) = σ ( a (cid:48) ) = σ ([ a (cid:48) ] ) ∈ { L, R } ,p < q, a ≤ B : σ ( a ) = σ ([ a ] ) = σ ( a (cid:48) ) = σ ([ a (cid:48) ] ) ∈ { L, R } ,p < q, C ≤ a : σ ( a ) = σ ( a (cid:48) ) ∈ { L, R } , σ ([ a ] ) = σ ([ a (cid:48) ] ) = U. If the arc a is a circle arc, we have σ ( a ) = σ ([ a ] ) = σ ( a (cid:48) ) = σ ([ a (cid:48) ] ) = U. HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 23
In each case, the contribution to both sides of equation (20) is the same. The claim for a type W3 saddle S : p n +1 ( r ) → p (cid:48) n − ( r ) now follows from reading off terms in ∂ P (cid:48) P ( S ). For a type W3 saddle S : p (cid:48) n − ( r ) → p n +1 ( r ), the map ∂ P P (cid:48) ( S ) : C P (cid:48) → C P is zero, thus proving the claim for this type of saddle.Next consider a type W4 saddle. We have C S = C P ⊕ C P (cid:48) , C P = C P = ( W ) n +2 ⊕ ( W ) n , C P (cid:48) = C P (cid:48) = ( W (cid:48) ) n ⊕ ( W (cid:48) ) n − . We can define vector spaces W and W (cid:48) such that C P = W ⊗ A , C P (cid:48) = W (cid:48) ⊗ A , and Q P (cid:48) P ( S ) : C P → C P (cid:48) is given by Q P (cid:48) P ( S ) = [ Q P (cid:48) P ] ⊗ A for a linear map [ Q P (cid:48) P ] : W → W (cid:48) . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving the one additional crossing of T , andlinear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles [ p n +1 ( r )] ↔ [ p (cid:48) n − ( r )] and[ p n +1 ( r )] ↔ [ p (cid:48) n − ( r )] induced by S : p n +1 ( r ) ↔ p (cid:48) n − ( r ): C [ P ] = ( W ) n +2 C [ P (cid:48) ] = ( W (cid:48) ) n C [ P ] = ( W ) n C [ P (cid:48) ] = ( W (cid:48) ) n − . Q [ P ]1[ P ]0 =1 W Q [ P (cid:48) ]0[ P ]0 =[ Q P (cid:48) P ] Q [ P (cid:48) ]1[ P (cid:48) ]0 =1 W (cid:48) Q [ P (cid:48) ]1[ P ]1 =[ Q P (cid:48) P ] (21)For a type W4 saddle S : p n +1 ( r ) → p (cid:48) n − ( r ), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is ∂ P (cid:48) P ( S ) = (cid:32) ˜ ∂ L [ P (cid:48) ] Q [ P (cid:48) ] [ P ] + Q [ P (cid:48) ] [ P ] ˜ ∂ L [ P ]
00 ˜ ∂ L [ P (cid:48) ] Q [ P (cid:48) ] [ P ] + Q [ P (cid:48) ] [ P ] ˜ ∂ L [ P ] (cid:33) . An argument similar to that used for type W3 saddles shows that ∂ P (cid:48) P ( S ) = ( ˜ ∂ LW (cid:48) [ Q P (cid:48) P ] + [ Q P (cid:48) P ] ˜ ∂ LW ) ⊗ A = ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP , thus proving the claim for this type of saddle. For a type W4 saddle S : p (cid:48) n − ( r ) → p n +1 ( r ), the map ∂ P P (cid:48) ( S ) : C P (cid:48) → C P is zero, thus proving the claim for this type of saddle. (cid:3) Reduction of type-changing saddles p n ( r ) ↔ p (cid:48) n ( r + 1)Recall from Section 5 that for each type-changing saddle S : p → p (cid:48) obtained from the resolution of T wedefined a vector space C S and a linear map ∂ S : C S → C S . The vector space C S is given by C S = C P ⊕ C P (cid:48) for P = τ ( p ) and P (cid:48) = τ ( p (cid:48) ). The linear map ∂ S is obtained by keeping only those terms of ∂ T + that arelinear maps or products of linear maps corresponding to saddles used to define ∂ P or ∂ P (cid:48) , or to saddles[ p ] → [ p (cid:48) ] or [ p ] → [ p (cid:48) ] induced by S : p → p (cid:48) .For a type-changing saddle p n ( r ) ↔ p (cid:48) n ( r + 1), Lemma 4.1 implies that the saddle arcs ( s , s ) in p n ( r )are such that τ ( s ) = τ ( s ) ∈ { e, u } or τ ( s ) = τ ( s ) = c . We classify saddles p n ( r ) ↔ p (cid:48) n ( r + 1) into thefollowing types, depending on the types and relative positions of the arcs s , s , B , and C of p n ( r ):Type C1: τ ( B ) = τ ( C ) ∈ { e, u } , τ ( s ) = τ ( s ) ∈ { e, u } , and either B < C ≤ s < s or C < B ≤ s < s ,Type C2: τ ( B ) = τ ( C ) ∈ { e, u } , τ ( s ) = τ ( s ) ∈ { e, u } , and either s ≤ B < C ≤ s or s ≤ C < B ≤ s ,Type C3: τ ( B ) = τ ( C ) ∈ { e, u } , τ ( s ) = τ ( s ) ∈ { e, u } , and either s < s ≤ B < C or s < s ≤ C < B ,Type C4: τ ( B ) = τ ( C ) ∈ { e, u } and τ ( s ) = τ ( s ) = c ,Type C5: τ ( B ) = τ ( C ) = c , and ( s , s ) do not lie on the circle component containing arcs B and C ,Type C6: τ ( B ) = τ ( C ) = c , and ( s , s ) lie on the circle component containing arcs B and C .For a saddle S : p → p (cid:48) , the linear map ∂ S has the form ∂ S = (cid:18) ∂ P ∂ P (cid:48) P ( S ) ∂ P (cid:48) (cid:19) for some linear map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) . For each type of saddle, we describe the residual terms andreduction factors obtained from ∂ P (cid:48) P ( S ). (Recall that in Lemma 6.5 we described the residual terms andreduction factors obtained from ∂ P and ∂ P (cid:48) .) For saddles p ( r ) ↔ p (cid:48) n ( r + 1) connecting a strand arc and a circle arc, we can define a vector space W (cid:48) such that ( C P (cid:48) , ∂ P (cid:48) ) = ( W (cid:48) ⊗ A, ∂ W (cid:48) ⊗ A + ˜ ∂ cW (cid:48) ⊗ x ) , where ˜ ∂ cW (cid:48) : W (cid:48) → W (cid:48) is the linear map corresponding to saddles connecting disk circles to circle arcs in P (cid:48) that lie in the additional circle component of P (cid:48) . We can express T P (cid:48) P ( S ) : C P → C P (cid:48) as T P (cid:48) P ( S ) = [ T P (cid:48) P ] ⊗ ˙ η for a linear map [ T P (cid:48) P ] : C P → W (cid:48) , and we can express T P P (cid:48) ( S ) : C P (cid:48) → C P as T P P (cid:48) ( S ) = [ T P P (cid:48) ] ⊗ ˙ (cid:15) for alinear map [ T P P (cid:48) ] : W (cid:48) → C P . Lemma 8.1.
For a type C1 saddle S : p n ( r ) → p (cid:48) n ( r + 1) we have the following terms. We have the residualterm T P (cid:48) P ( S ) : C P → C P (cid:48) . For
B < C , reduction on P n yields the reduction factor β ( P, S ) = ˜ T LP (cid:48) P ( S ) + ˜ T RP (cid:48) P ( S ) : C P → C P (cid:48) . For
C < B , reduction on P (cid:48) n yields the reduction factor α ( P (cid:48) , S ) = ˜ T LP (cid:48) P ( S ) + ˜ T RP (cid:48) P ( S ) : C P → C P (cid:48) . For a type C1 saddle S : p (cid:48) n ( r + 1) → p n ( r ) we have the following terms. We have the residual term T P P (cid:48) ( S ) : C P (cid:48) → C P . For
B < C , reduction on P (cid:48) n yields the reduction factor β ( P (cid:48) , S ) = ˜ T LP P (cid:48) ( S ) + ˜ T RP P (cid:48) ( S ) : C P (cid:48) → C P . For
C < B , reduction on P n yields the reduction factor α ( P, S ) = ˜ T LP P (cid:48) ( S ) + ˜ T RP P (cid:48) ( S ) : C P (cid:48) → C P . Proof.
We will prove the claim for the case
B < C ; the case
C < B is similar. We have C S = C P ⊕ C P (cid:48) , C P = ( C P ⊗ A ) n +1 ⊕ ( C P ) n +1 , C P (cid:48) = ( C P (cid:48) ⊗ A ) n +1 ⊕ ( C P (cid:48) ) n +1 . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles[ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] and [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] induced by S : p n ( r ) ↔ p (cid:48) n ( r + 1): C [ P ] = ( C P ⊗ A ) n +1 C [ P (cid:48) ] = ( C P (cid:48) ⊗ A ) n +1 C [ P ] = ( C P ) n +1 C [ P (cid:48) ] = ( C P (cid:48) ) n +1 . T [ P ]1[ P ]0 =1 CP ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P ]0 =1 CP ⊗ (cid:15) T [ P (cid:48) ]0[ P ]0 =[ T P (cid:48) P ] ⊗ ∆ T [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ ˙ (cid:15) ˜ T L [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ (cid:15)T [ P ]0[ P (cid:48) ]0 =[ T PP (cid:48) ] ⊗ mT [ P (cid:48) ]1[ P ]1 =[ T P (cid:48) P ] ⊗ ˙ η ˜ T L [ P (cid:48) ]1[ P ]1 =[ T RP (cid:48) P ] ⊗ ηT [ P ]1[ P (cid:48) ]1 =[ T PP (cid:48) ] ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P (cid:48) ]1 =[ T RPP (cid:48) ] ⊗ (cid:15) (22)For a saddle S : p n ( r ) → p (cid:48) n ( r + 1), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is given by ∂ P (cid:48) P ( S ) = (cid:18) [ T P (cid:48) P ] ⊗ ∆ 00 [ T P (cid:48) P ] ⊗ ˙ η (cid:19) . The claim for a saddle S : p n ( r ) → p (cid:48) n ( r + 1) now follows from reading off terms in ∂ P (cid:48) P ( S ) and using thefact that ∆ = ˙ η ⊗ A + η ⊗ x . The case of a saddle S : p (cid:48) n ( r + 1) → p n ( r ) is similar. (cid:3) Lemma 8.2.
For a type C2 saddle S : p n ( r ) → p (cid:48) n ( r + 1) we have the following terms. We have the residualterm T P (cid:48) P ( S ) : C P → C P (cid:48) . For
B < C , reduction on P n yields the reduction factor β ( P, S ) = ˜ T LP (cid:48) P ( S ) + ˜ T RP (cid:48) P ( S ) : C P → C P (cid:48) . HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 25
For a type C2 saddle S : p (cid:48) n ( r + 1) → p n ( r ) we have the following terms. We have the residual term T P P (cid:48) ( S ) : C P (cid:48) → C P . For
C < B , reduction on P n yields the reduction factor α ( P, S ) = ˜ T LP P (cid:48) ( S ) + ˜ T RP P (cid:48) ( S ) : C P (cid:48) → C P . Proof.
We will prove the claim for the case
B < C ; the case
C < B is similar. We have C S = C P ⊕ C P (cid:48) , C P = ( C P ⊗ A ) n +1 ⊕ ( C P ) n +1 , C P (cid:48) = ( W (cid:48) ) n +1 ⊕ ( W (cid:48) ) n − . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles[ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] and [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] induced by S : p n ( r ) ↔ p (cid:48) n ( r + 1): C [ P ] = ( C P ⊗ A ) n +1 C [ P (cid:48) ] = ( W (cid:48) ) n +1 C [ P ] = ( C P ) n +1 C [ P (cid:48) ] = ( W (cid:48) ) n − . T [ P ]1[ P ]0 =1 CP ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P ]0 =1 CP ⊗ (cid:15) T [ P (cid:48) ]0[ P ]0 =[ T P (cid:48) P ] ⊗ ˙ (cid:15) ˜ T L [ P (cid:48) ]0[ P ]0 =[ T RP (cid:48) P ] ⊗ (cid:15) Q [ P (cid:48) ]1[ P (cid:48) ]0 =1 W (cid:48) T [ P ]0[ P (cid:48) ]0 =[ T PP (cid:48) ] ⊗ ˙ η ˜ T L [ P ]0[ P (cid:48) ]0 =[ T RPP (cid:48) ] ⊗ ηQ [ P (cid:48) ]1[ P ]1 =[ T P (cid:48) P ] (23)For a saddle S : p n ( r ) → p (cid:48) n ( r + 1), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is given by ∂ P (cid:48) P ( S ) = (cid:18) [ T P (cid:48) P ] ⊗ ˙ (cid:15) T P (cid:48) P ] ⊗ (cid:15) ˜ ∂ L [ P (cid:48) ] [ T P (cid:48) P ] + [ T P (cid:48) P ] ˜ ∂ L [ P ] (cid:19) . The matrix element [ T P (cid:48) P ] ⊗ (cid:15) : ( C P ⊗ A ) n +1 → ( W (cid:48) ) n − is obtained from the interleaved saddles ( S :[ p n ( r )] → [ p (cid:48) n ( r + 1)] , T : [ p (cid:48) n ( r + 1)] → [ p (cid:48) n ( r + 1)] ). The claim for a saddle S : p n ( r ) → p (cid:48) n ( r + 1) nowfollows from reading off terms in ∂ P (cid:48) P ( S ). For a saddle S : p (cid:48) n ( r + 1) → p n ( r ), the map ∂ P P (cid:48) ( S ) : C P (cid:48) → C P is given by ∂ P P (cid:48) ( S ) = (cid:18) [ T P P (cid:48) ] ⊗ ˙ η
00 0 (cid:19) . The claim for a saddle S : p (cid:48) n ( r + 1) → p n ( r ) now follows from reading off terms in ∂ P P (cid:48) ( S ). (cid:3) Lemma 8.3.
For a type C3, C4, C5, or C6 saddle S : p n ( r ) → p (cid:48) n ( r + 1) , we have the following terms.There are no reduction factors. We have the residual term T P (cid:48) P ( S ) : C P → C P (cid:48) . For a type C3, C4, C5, or C6 saddle S : p (cid:48) n ( r + 1) → p n ( r ) , we have the following terms. There are noreduction factors. We have the residual term T P P (cid:48) ( S ) : C P (cid:48) → C P . Proof.
First consider a type C3 saddle. We will prove the claim for the case
B < C ; the case
C < B issimilar. We have C S = C P ⊕ C P (cid:48) , C P = ( C P ⊗ A ) n +1 ⊕ ( C P ) n +1 , C P (cid:48) = ( C P (cid:48) ⊗ A ) n +1 ⊕ ( C P (cid:48) ) n +1 . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] and [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] induced by S : p n ( r ) ↔ p (cid:48) n ( r + 1): C [ P ] = ( C P ⊗ A ) n +1 C [ P (cid:48) ] = ( C P (cid:48) ⊗ A ) n +1 C [ P ] = ( C P ) n +1 C [ P (cid:48) ] = ( C P (cid:48) ) n +1 . T [ P ]1[ P ]0 =1 CP ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P ]0 =1 CP ⊗ (cid:15) T [ P (cid:48) ]0[ P ]0 =[ T P (cid:48) P ] ⊗ ˙ η ⊗ A ˜ T L [ P (cid:48) ]0[ P ]0 =[ T LP (cid:48) P ] ⊗ η ⊗ A T [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ ˙ (cid:15) ˜ T L [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ (cid:15)T [ P ]0[ P (cid:48) ]0 =[ T PP (cid:48) ] ⊗ ˙ (cid:15) ⊗ A ˜ T L [ P ]0[ P (cid:48) ]0 =[ T LPP (cid:48) ] ⊗ (cid:15) ⊗ A T [ P (cid:48) ]1[ P ]1 =[ T P (cid:48) P ] ⊗ ˙ η ˜ T L [ P (cid:48) ]1[ P ]1 =[ T LP (cid:48) P ] ⊗ ηT [ P ]1[ P (cid:48) ]1 =[ T PP (cid:48) ] ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P (cid:48) ]1 =[ T LPP (cid:48) ] ⊗ (cid:15) (24)For a type C3 saddle S : p n ( r ) → p (cid:48) n ( r + 1), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is given by ∂ P (cid:48) P ( S ) = (cid:18) [ T P (cid:48) P ] ⊗ ˙ η ⊗ A
00 [ T P (cid:48) P ] ⊗ ˙ η (cid:19) . The claim for type C3 saddles now follows from reading off terms in ∂ P (cid:48) P ( S ). The case of a type C3 saddle S : p (cid:48) n ( r + 1) → p n ( r ) is similar.Next consider a type C4 saddle. We will prove the claim for the case B < C ; the case
C < B is similar.A circle component in P that does not contain the arcs B and C is split into two circle components in P (cid:48) , so we can define vector spaces W and W (cid:48) such that C P = W ⊗ A and C P (cid:48) = W (cid:48) ⊗ A ⊗ A . We canexpress T P (cid:48) P ( S ) as T P (cid:48) P ( S ) = [ T P (cid:48) P ] ⊗ ∆ for a linear map [ T P (cid:48) P ] : W → W (cid:48) , and we can express T P P (cid:48) ( S )as T P P (cid:48) ( S ) = [ T P P (cid:48) ] ⊗ m for a linear map [ T P P (cid:48) ] : W (cid:48) → W . We have C S = C P ⊕ C P (cid:48) , C P = ( C P ⊗ A ) n +1 ⊕ ( C P ) n +1 , C P (cid:48) = ( C P (cid:48) ⊗ A ) n +1 ⊕ ( C P (cid:48) ) n +1 . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles[ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] and [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] induced by S : p n ( r ) ↔ p (cid:48) n ( r + 1): C [ P ] = ( C P ⊗ A ) n +1 C [ P (cid:48) ] = ( C P (cid:48) ⊗ A ) n +1 C [ P ] = ( C P ) n +1 C [ P (cid:48) ] = ( C P (cid:48) ) n +1 . T [ P ]1[ P ]0 =1 CP ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P ]0 =1 CP ⊗ (cid:15) T [ P (cid:48) ]0[ P ]0 =[ T P (cid:48) P ] ⊗ ∆ ⊗ A T [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ ˙ (cid:15) ˜ T L [ P (cid:48) ]1[ P (cid:48) ]0 =1 CP (cid:48) ⊗ (cid:15)T [ P ]0[ P (cid:48) ]0 =[ T PP (cid:48) ] ⊗ m ⊗ A T [ P (cid:48) ]1[ P ]1 =[ T P (cid:48) P ] ⊗ ∆ T [ P ]1[ P (cid:48) ]1 =[ T PP (cid:48) ] ⊗ m (25)For a type C4 saddle S : p n ( r ) → p (cid:48) n ( r + 1), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is given by ∂ P (cid:48) P ( S ) (cid:18) [ T P (cid:48) P ] ⊗ ∆ ⊗ A
00 [ T P (cid:48) P ] ⊗ ∆ (cid:19) . The claim for type C4 saddles now follows from reading off terms in ∂ P (cid:48) P ( S ). The case of a type C4 saddle S : p (cid:48) n ( r + 1) → p n ( r ) is similar.We claim that for a type C5 or C6 saddle S : p ( r ) → p (cid:48) ( r + 1) we have C S = C P ⊕ C P (cid:48) , C P = C P , C P (cid:48) = C P (cid:48) , ∂ P (cid:48) P ( S ) = T P (cid:48) P ( S ) . Consider a type C5 saddle. We can define vector spaces W and W (cid:48) such that C P = W ⊗ A and C P (cid:48) = W (cid:48) ⊗ A .We can express T P (cid:48) P ( S ) as T P (cid:48) P ( S ) = [ T P (cid:48) P ] ⊗ A for a linear map [ T P (cid:48) P ] : W → W (cid:48) , and we can express T P P (cid:48) ( S ) as T P P (cid:48) ( S ) = [ T P P (cid:48) ] ⊗ A for a linear map [ T P P (cid:48) ] : W (cid:48) → W . We have C P = C P = ( W ) n +1 ⊕ ( W ) n − , C P (cid:48) = C P (cid:48) = ( W (cid:48) ) n +1 ⊕ ( W (cid:48) ) n − . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 27 [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] and [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] induced by S : p n ( r ) ↔ p (cid:48) n ( r + 1): C [ P ] = ( W ) n +1 C [ P (cid:48) ] = ( W (cid:48) ) n +1 C [ P ] = ( W ) n − C [ P (cid:48) ] = ( W (cid:48) ) n − . Q [ P ]1[ P ]0 =1 W T [ P (cid:48) ]0[ P ]0 =[ T P (cid:48) P ]˜ T L [ P (cid:48) ]0[ P ]0 =[ ˜ T LP (cid:48) P ] Q [ P (cid:48) ]1[ P (cid:48) ]0 =1 W (cid:48) T [ P ]0[ P (cid:48) ]0 =[ T PP (cid:48) ]˜ T L [ P ]0[ P (cid:48) ]0 =[ ˜ T LPP (cid:48) ] T [ P (cid:48) ]1[ P ]1 =[ T P (cid:48) P ]˜ T L [ P (cid:48) ]1[ P ]1 =[ ˜ T LP (cid:48) P ] T [ P ]1[ P (cid:48) ]1 =[ T PP (cid:48) ]˜ T L [ P ]1[ P (cid:48) ]1 =[ ˜ T LPP (cid:48) ] (26)For a type C5 saddle S : p n ( r ) → p (cid:48) n ( r + 1), the map ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is given by ∂ P (cid:48) P ( S ) = (cid:18) [ T P (cid:48) P ] 00 [ T P (cid:48) P ] (cid:19) = [ T P (cid:48) P ] ⊗ A = T P (cid:48) P ( S ) . The case of a type C5 saddle S : p (cid:48) n ( r + 1) → p n ( r ) is similar.Consider a type C6 saddle. A circle component in P that contains the arcs B and C is split into two circlecomponents in P (cid:48) , so we can define vector spaces W and W (cid:48) such that C P = W ⊗ A and C P (cid:48) = W (cid:48) ⊗ A ⊗ A .We can express T P (cid:48) P ( S ) as T P (cid:48) P ( S ) = [ T P (cid:48) P ] ⊗ ∆ for a linear map [ T P (cid:48) P ] : W → W (cid:48) , and we can express T P P (cid:48) ( S ) as T P P (cid:48) ( S ) = [ T P P (cid:48) ] ⊗ m for a linear map [ T P P (cid:48) ] : W (cid:48) → W . We have C S = C P ⊕ C P (cid:48) , C P = C P = ( W ) n +1 ⊕ ( W ) n − , C P (cid:48) = C P (cid:48) = ( W (cid:48) ⊗ A ) n +1 ⊕ ( W (cid:48) ⊗ A ) n − . We have linear maps C [ P ] → C [ P ] and C [ P (cid:48) ] → C [ P (cid:48) ] corresponding to saddles obtained by resolving theone additional crossing of T , and linear maps C [ P ] ↔ C [ P (cid:48) ] and C [ P ] ↔ C [ P (cid:48) ] corresponding to saddles[ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] and [ p n ( r )] ↔ [ p (cid:48) n ( r + 1)] induced by S : p n ( r ) ↔ p (cid:48) n ( r + 1): C [ P ] = ( W ) n +1 C [ P (cid:48) ] = ( W (cid:48) ⊗ A ) n +1 C [ P ] = ( W ) n − C [ P (cid:48) ] = ( W (cid:48) ⊗ A ) n − . Q [ P ]1[ P ]0 =1 W T [ P (cid:48) ]0[ P ]0 =[ T P (cid:48) P ] ⊗ ˙ η ˜ T L [ P (cid:48) ]0[ P ]0 =[ T σP (cid:48) P ] ⊗ η Q [ P (cid:48) ]1[ P (cid:48) ]0 =1 W (cid:48) ⊗ A T [ P ]0[ P (cid:48) ]0 =[ T PP (cid:48) ] ⊗ ˙ (cid:15) ˜ T L [ P ]0[ P (cid:48) ]0 =[ T σPP (cid:48) ] ⊗ (cid:15)T [ P (cid:48) ]1[ P ]1 =[ T P (cid:48) P ] ⊗ ˙ η ˜ T L [ P (cid:48) ]1[ P ]1 =[ T σP (cid:48) P ] ⊗ ηT [ P ]1[ P (cid:48) ]1 =[ T PP (cid:48) ] ⊗ ˙ (cid:15) ˜ T L [ P ]1[ P (cid:48) ]1 =[ T σPP (cid:48) ] ⊗ (cid:15) (27)The orientation σ ∈ { L, R } depends on the particular saddle. For a type C6 saddle S : p ( r ) → p (cid:48) ( r + 1), themap ∂ P (cid:48) P ( S ) : C P → C P (cid:48) is given by ∂ P (cid:48) P ( S ) = (cid:18) [ T P (cid:48) P ] ⊗ ˙ η T P (cid:48) P ] ⊗ η [ T P (cid:48) P ] ⊗ ˙ η (cid:19) = [ T P (cid:48) P ] ⊗ ˙ η ⊗ A + [ T P (cid:48) P ] ⊗ η ⊗ x = [ T P (cid:48) P ] ⊗ ∆ = T P (cid:48) P ( S ) . The matrix element [ T P (cid:48) P ] ⊗ η : ( W ) n +1 → ( W (cid:48) ⊗ A ) n − is obtained from the nested saddles ( S : [ p n ( r )] → [ p n ( r )] , T : [ p n ( r )] → [ p (cid:48) n ( r + 1)] ). The case of a type C6 saddle S : p (cid:48) ( r + 1) → p ( r ) is similar. (cid:3) All terms of ∂ T + occur in ∂ red We are now ready to complete the proof of Theorem 3.1. Recall that in Definition 6.4 we defined thereduction ( C red , ∂ red ) of the chain complex ( C T + , ∂ T + ). In Lemma 6.8 we showed that C red = C T + asbigraded vector spaces. It remains to show that ∂ red = ∂ T + . In this section we show that all the termsof ∂ T + occur in ∂ red , and in Section 10 we show that all the terms of ∂ red occur in ∂ T + . Recall that ∂ T + = ∂ T + + ∂ + T + , where ∂ T + = (cid:88) P ∂ P + (cid:88) P (cid:88) P (cid:48) (cid:54) = P T P (cid:48) P , ∂ + T + = (cid:88) P (cid:88) P (cid:48) (cid:88) P (cid:48)(cid:48) ( ˜ T LP (cid:48)(cid:48) P (cid:48) Q P (cid:48) P + Q P (cid:48)(cid:48) P (cid:48) ˜ T LP (cid:48) P ) . Lemma 9.1.
All the terms of ∂ T + occur as residual terms of ∂ red . Proof.
In Lemma 6.5 we showed that for each tangle type P we obtain ∂ P as a residual term. For P (cid:48) (cid:54) = P ,we have T P (cid:48) P = (cid:88) S T P (cid:48) P ( S ) , where the sum is taken over all type-changing saddles S : p n ( r ) → p (cid:48) n ( r ±
1) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) .In Section 8 we showed that for each type-changing saddle S : p n ( r ) → p (cid:48) n ( r ±
1) we obtain the residual term T P (cid:48) P ( S ) : C P → C P (cid:48) . (cid:3) Lemma 9.2.
All the terms of ∂ + T + occur as residual or reduction terms of ∂ red .Proof. We can express ∂ + T + as ∂ + T + = (cid:88) P (cid:88) P (cid:88) P (cid:48)(cid:48) ( ˜ T LP (cid:48)(cid:48) P Q P P + Q P (cid:48)(cid:48) P ˜ T LP P ) = (cid:88) P (cid:88) P (cid:48)(cid:48) ∂ P (cid:48)(cid:48) P , where ∂ P (cid:48)(cid:48) P = (cid:88) P ( ˜ T LP (cid:48)(cid:48) P Q P P + Q P (cid:48)(cid:48) P ˜ T LP P ) . We can express ∂ P (cid:48)(cid:48) P as ∂ P (cid:48)(cid:48) P = ∂ = P (cid:48)(cid:48) P + ∂ (cid:54) = P (cid:48)(cid:48) P , where ∂ = P (cid:48)(cid:48) P = ˜ ∂ LP (cid:48)(cid:48) Q P (cid:48)(cid:48) P + Q P (cid:48)(cid:48) P ˜ ∂ LP , ∂ (cid:54) = P (cid:48)(cid:48) P = (cid:88) P (cid:54) = P (cid:48)(cid:48) ˜ T LP (cid:48)(cid:48) P Q P P + (cid:88) P (cid:54) = P Q P (cid:48)(cid:48) P ˜ T LP P . Lemma 9.4 below shows that each term of ∂ = P (cid:48)(cid:48) P is a term of ∂ red .As discussed in Remark 5.3, the map ∂ (cid:54) = P (cid:48)(cid:48) P : C P → C P (cid:48)(cid:48) is a sum of products of pairs of maps correspondingto pairs of saddles obtained from the resolution of T . The pairs of saddles whose corresponding maps areincluded in the sum comprise two adjacent sides of a commuting square of saddles corresponding to interleavedsaddles ( S, T ) of the forms shown in diagram (7) and nested saddles (
S, T ) of the forms shown in diagrams(8) and (9), where τ ( p ) = P , τ ( p (cid:48)(cid:48) ) = P (cid:48)(cid:48) , and all saddles in the commuting square change the tangle type.We claim that the contribution of each such pair occurs in ∂ red . We will show this for a few representativeexamples.For the first form of interleaved saddles ( S, T ) shown in diagram (7), the contribution to ∂ (cid:54) = P (cid:48)(cid:48) P is˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + ˜ T LP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) . Lemma 9.5 below shows that this contribution occurs in ∂ red . For the first form of nested saddles ( S, T )shown in diagram (8), the contribution to ∂ (cid:54) = P (cid:48)(cid:48) P is˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) + Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T L ˜ P P ( T ) . Lemma 9.6 below shows that this contribution occurs in ∂ red . (cid:3) Lemmas 9.1 and 9.2 give
Lemma 9.3.
All the terms of ∂ T + occur as residual or reduction terms of ∂ red . We now complete the proof of Lemma 9.2:
Lemma 9.4.
For each saddle S : p n +1 ( r ) → p (cid:48) n − ( r ) with τ ( p ) = P and τ ( p (cid:48) ) = P (cid:48) , we obtain a reductionor residual term ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP : C P → C P (cid:48) . Proof.
For saddles S : p n +1 ( r ) → p (cid:48) n − ( r ) of type W2, Lemmas 6.5 and 7.3 show that we obtain the term˜ ∂ c P (cid:48) Q P (cid:48) P ( S ) from the reduction on P (cid:48) and the term Q P (cid:48) P ( S ) ˜ ∂ c P from the reduction on P . We claim˜ ∂ c P (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ c P = ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP . (28)We can see this as follows. Let ( p , p ) denote the saddle insertion points of S in p n +1 . Consider a saddleconnecting a disk circle to an arc a of p n +1 with saddle insertion point q ∈ a . There is a correspondingsaddle connecting the disk circle to an arc a (cid:48) of p (cid:48) n − with saddle insertion point q (cid:48) ∈ a (cid:48) . The arc a of p n +1 lies on a unique arc or disk circle [ a ] of [ p n +1 ] and [ a ] of [ p n +1 ] . The arc a (cid:48) of p (cid:48) n − lies on a unique arc HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 29 or disk circle [ a (cid:48) ] of [ p (cid:48) n − ] and [ a (cid:48) ] of [ p (cid:48) n − ] . If a is a strand arc, the orientations and types of these arcsare as follows: q < p : σ ( a ) = σ ( a (cid:48) ) ∈ { L, R } , τ ([ a ] ) ∈ { e, u } , τ ([ a (cid:48) ] ) ∈ { e, u } ,p < q < p , a ≤ C : σ ( a ) = σ ( a (cid:48) ) ∈ { L, R } , τ ([ a ] ) ∈ { e, u } , τ ([ a (cid:48) ] ) = c ,p < q < p , B ≤ a : σ ( a ) = σ ( a (cid:48) ) ∈ { L, R } , τ ([ a ] ) = c , τ ([ a (cid:48) ] ) ∈ { e, u } ,p < q : σ ( a ) = σ ( a (cid:48) ) ∈ { L, R } , τ ([ a ] ) = c , τ ([ a (cid:48) ] ) = c , where c and c indicate circle arcs on the additional circle components of [ p (cid:48) ] and [ p ] , respectively. If a isa circle arc, we have τ ( a ) = τ ([ a ] ) = τ ( a (cid:48) ) = τ ([ a (cid:48) ] ) = c ;note that the circle arc [ a ] in [ p ] does not lie on additional circle component of [ p ] , and the circle arc [ a (cid:48) ] in [ p (cid:48) ] does not lie on the additional circle component of [ p (cid:48) ] . In each case, the contribution to both sidesof equation (28) is the same.For saddles S : p n +1 ( r ) → p (cid:48) n − ( r ) of type W3 or W4, Lemma 7.4 shows that we obtain ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP as a residual term. Thus the claim holds for saddles S : p n +1 ( r ) → p (cid:48) n − ( r ) of all possibletypes. (cid:3) We classify interleavings ( S : p n ( r ) → p (cid:48) n − ( r ) , T : p n ( r ) → ˜ p n − ( r )) into the following types, dependingon the types and relative positions of the arcs B and C , the saddle arcs ( s , s ) of S , and the saddle arcs( t , t ) of T in p n ( r ):Type I1: τ ( B ) = τ ( C ) ∈ { e, u } and s ≤ t ≤ C < B ≤ s ≤ t ,Type I2: τ ( B ) = τ ( C ) ∈ { e, u } and s ≤ t ≤ s ≤ C < B ≤ t ,Type I3: τ ( B ) = τ ( C ) ∈ { e, u } and either s ≤ t ≤ s ≤ t ≤ B < C or s ≤ t ≤ s ≤ t ≤ C < B ,Type I4: τ ( B ) = τ ( C ) = c .In the classification we have made use of Lemma 7.1, which states that type W1 saddles do not actuallyoccur, and Lemma 7.2, which states that type W2 saddles must have C < B . Lemma 9.5.
Consider interleaved saddles ( S : p n ( r ) → p (cid:48) n − ( r ) , T : p n ( r ) → ˜ p n − ( r )) and induced saddles T : p (cid:48) n − ( r ) → p (cid:48)(cid:48) n − ( r + 1) and S : ˜ p n − ( r ) → p (cid:48)(cid:48) n − ( r + 1) . We obtain the term ˜ T LP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) = ˜ T RP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) : C P → C P (cid:48)(cid:48) , and we have ˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = ˜ T RP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) = 0 . Proof.
The interleaved saddles (
S, T ) are described by the commutative diagram p n ( r ) p (cid:48) n − ( r )˜ p n − ( r ) p (cid:48)(cid:48) n − ( r + 1) . ST TS
We have the following cases:
I S T T S reduction term β ( T ) α ( S )? reduction term β ( S ) α ( T )? residual term?I1 W2 C1 W2 C3 y n n I2 W3 C2 W2 C2 n y n
I3 W3 C3 W3 C3 n n y
I4 W4 C5 W4 C5 n n y
The column marked I indicates the type of interleaving of the pair of saddles ( S, T ), the columns marked S , T , T , and S indicate the corresponding types of each of these saddles, and the remaining three columns indicatewhether we obtain ˜ T LP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) = ˜ T RP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) as a reduction term β ( T ) α ( S ), as a reductionterm β ( S ) α ( T ), or as a residual term. The entries for reduction terms follow from the enumeration of such terms in the proof of Lemma 10.3, and the entries for residual terms follow from the enumeration of suchterms in the proof of Lemma 10.1. The fact that ˜ T LP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = ˜ T RP (cid:48)(cid:48) ˜ P ( S ) Q ˜ P P ( T ) = 0 was shown inthe proof of Lemma 5.4. (cid:3) We classify nestings ( S : p n ( r ) → p (cid:48) n − ( r ) , T : p n ( r ) → ˜ p n ( r + 1)) into the following types, depending onthe types and relative positions of the arcs B and C , the saddle arcs ( s , s ) of S , and the saddle arcs ( t , t )of T in p n ( r ):Type N1: τ ( B ) = τ ( C ) ∈ { e, u } and s ≤ C < B ≤ t < t ≤ s ,Type N2: τ ( B ) = τ ( C ) ∈ { e, u } and s ≤ t ≤ C < B ≤ t ≤ s ,Type N3: τ ( B ) = τ ( C ) ∈ { e, u } and s ≤ t < t ≤ C < B ≤ s ,Type N4: τ ( B ) = τ ( C ) ∈ { e, u } and either s ≤ t < t ≤ s ≤ B < C or s ≤ t < t ≤ s ≤ C < B ,Type N5: τ ( B ) = τ ( C ) = c .In the classification we have made use of Lemma 7.1, which states that type W1 saddles do not actuallyoccur, and Lemma 7.2, which states that type W2 saddles must have C < B . Lemma 9.6.
Consider nested saddles ( S : p n ( r ) → p (cid:48) n − ( r ) , T : p n ( r ) → ˜ p n ( r + 1)) with induced saddles T : p (cid:48) n − ( r ) → p (cid:48)(cid:48) n − ( r + 1) and S : ˜ p n ( r + 1) → p (cid:48)(cid:48) n − ( r + 1) . For each such pair of saddles, we obtain a term ˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T σ ˜ P P ( T ) : C P → C P (cid:48)(cid:48) for some orientation σ ∈ { L, R } , and we have Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T σ ˜ P P ( T ) = ˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = 0 . Proof.
The nested saddles (
S, T ) are described by the commutative diagram p n ( r ) p (cid:48) n − ( r )˜ p n ( r + 1) p (cid:48)(cid:48) n − ( r + 1) . ST TS
We have the following cases:
N S T T S reduction term β ( T ) α ( S )? reduction term β ( S ) α ( T )? residual term?N1 W2 C3 C1 W2 n y n N2 W2 C2 C2 W4 y n n
N3 W2 C1 C3 W2 y n n
N4 W3 C3 C3 W3 n n y
N5 W4 C5 C5 W4 n n y
The column marked N indicates the type of nesting of the pair of saddles ( S, T ), the columns marked S , T , T , and S indicate the corresponding types of each of these saddles, and the remaining three columns indicatewhether we obtain ˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = Q P (cid:48)(cid:48) P ( S ) ˜ T σ ˜ P P ( T ) as a reduction term β ( T ) α ( S ), as a reductionterm β ( S ) α ( T ), or as a residual term. The entries for reduction terms follow from the enumeration of suchterms in the proof of Lemma 10.3, and the entries for residual terms follow from the enumeration of suchterms in the proof of Lemma 10.1. The fact that Q P (cid:48)(cid:48) ˜ P ( S ) ˜ T σ ˜ P P ( T ) = ˜ T σP (cid:48)(cid:48) P (cid:48) ( T ) Q P (cid:48) P ( S ) = 0 was shown inthe proof of Lemma 5.4. (cid:3) All terms of ∂ red occur in ∂ T + Recall that in Definition 6.4 we defined the reduction ( C red , ∂ red ) of the chain complex ( C T + , ∂ T + ). InSection 9 we showed that all terms of ∂ T + occur in ∂ red . In this section we show that all terms of ∂ red occurin ∂ T + , thus completing the proof of Theorem 3.1.Recall that ∂ T + = ∂ T + + ∂ + T + . The map ∂ T + is a sum of maps corresponding to saddles obtained fromthe resolution of T . The map ∂ + T + is a sum of products of maps corresponding to pairs of saddles obtainedfrom the resolution of T . As explained in Section 5, we classify these saddles into three types: HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 31 (1) Saddles [ p ] → [ p ] obtained by resolving the one additional crossing of T .(2) Saddles [ p ] → [ p (cid:48) ] and [ p ] → [ p (cid:48) ] that are induced by a type-preserving (i.e. τ ( p ) = τ ( p (cid:48) )) saddle S : p → p (cid:48) obtained from the resolution of T .(3) Saddles [ p ] → [ p (cid:48) ] and [ p ] → [ p (cid:48) ] that are induced by a type-changing (i.e. τ ( p ) (cid:54) = τ ( p (cid:48) )) saddle S : p → p (cid:48) obtained from the resolution of T .In Sections 6, 7, and 8, we described the residual terms and reduction factors that are obtained from termsof ∂ T + that correspond to saddles of all three types (1), (2), and (3), and from terms of ∂ + T + that correspondto pairs of saddles that are not both of type (3). It remains to describe the additional residual terms andreduction factors obtained from terms of ∂ + T + that correspond to pairs of saddles that are both of type (3). Lemma 10.1.
All the residual terms obtained from ∂ T + occur in ∂ T + .Proof. In Section 6 we showed that terms of ∂ T + corresponding to saddles of types (1) and (2) yield theresidual term (cid:88) P ∂ P . In Sections 7 and 8 we showed that if we now include saddles of type (3), but exclude terms of ∂ + T + corre-sponding to pairs of saddles that are both of type (3), we obtain the additional residual term (cid:88) P (cid:88) P (cid:48) (cid:54) = P T P (cid:48) P + (cid:88) S (cid:88) P (cid:88) P (cid:48) ( ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP ) , where the sum on S is taken over all saddles S : p n +1 ( r ) → p (cid:48) n − ( r ) of type W3 or W4. All of these residualterms occur in ∂ + T + .It remains to consider terms of ∂ + T + that correspond to pairs of saddles that are both of type (3). Suchterms take one of the following forms:˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] Q [ P (cid:48) ] [ P ] , ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] Q [ P (cid:48) ] [ P ] , Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] , Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] , where, for example, Q [ P (cid:48) ] [ P ] and ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] are maps corresponding to type (3) saddles [ p ] → [ p (cid:48) ] and[ p (cid:48) ] → [ p (cid:48)(cid:48) ] that occur in the resolution of T , which are induced by type-changing saddles S : p → p (cid:48) and S : p (cid:48) → p (cid:48)(cid:48) that occur in the resolution of T . In general, such a residual term has the form ∂ P (cid:48)(cid:48) P ( S , S ) = ∂ t ( S ) ∂ i ( S ) : C P → C P (cid:48)(cid:48) , where ∂ i ( S ) : C P → C T + and ∂ t ( S ) : C T + → C P (cid:48)(cid:48) are linear maps corresponding to saddles induced by S and S . We will refer to the maps ∂ i ( S ) and ∂ t ( S ) as the initial factor and terminal factor of theresidual term ∂ P (cid:48)(cid:48) P ( S , S ). For tangle types P of type P1A, we identify C P with the subspace C P ⊗ x of C P = ( C P ⊗ A ) ⊕ C P , so we have initial and terminal factors ∂ i ( S ) : C P ⊗ x → C T + , ∂ t ( S ) : C T + → C P ⊗ x. For tangle types P of type P1B, we identify C P with the subspace C P ⊗ A of C P = C P ⊕ ( C P ⊗ A ), so wehave initial and terminal factors ∂ i ( S ) : C P ⊗ A → C T + , ∂ t ( S ) : C T + → C P ⊗ A . For tangle types P of type P2, we have C P = C P = W ⊗ A for a vector space W , so we have initial andterminal factors ∂ i ( S ) : W → C T + , ∂ t ( S ) : C T + → W. Using the results of Sections 7 and 8, for each type-changing saddle S : p ↔ p (cid:48) obtained from the resolutionof T we indicate the possible initial and terminal factors that could be obtained:For a type W2 saddle S : p n +1 ( r ) ↔ p (cid:48) n − ( r ):P1B P1A (cid:48) . ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] W2 (29) For a type W3 saddle S : p n +1 ( r ) ↔ p (cid:48) n − ( r ):P1A P1A (cid:48) ,∂ i ( S ) = Q [ P (cid:48) ] [ P ] ∂ t ( S ) = Q [ P (cid:48) ] [ P ] W3 P1B P1B (cid:48) . ∂ i ( S ) = Q [ P (cid:48) ] [ P ] ∂ t ( S ) = Q [ P (cid:48) ] [ P ] W3 For a type W4 saddle S : p n +1 ( r ) ↔ p (cid:48) n − ( r ):P2 P2 (cid:48) .∂ i ( S ) = Q [ P (cid:48) ] [ P ] ∂ t ( S ) = Q [ P (cid:48) ] [ P ] ∂ i ( S ) = Q [ P (cid:48) ] [ P ] ∂ t ( S ) = Q [ P (cid:48) ] [ P ] W4 For a type C2 saddle S : p n ( r ) ↔ p (cid:48) n ( r + 1)P1A P2 (cid:48) ,∂ i ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = Q [ P (cid:48) ] [ P ] C2 P1A P2 (cid:48) , ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] C2 P1B P2 (cid:48) , ∂ t ( S ) = ˜ T L [ P (cid:48) ] [ P ] C2 P1B P2 (cid:48) . ∂ i ( S ) = Q [ P ] [ P (cid:48) ] ∂ t ( S ) = ˜ T L [ P ] [ P (cid:48) ] ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] C2 For a type C3 saddle S : p n ( r ) ↔ p (cid:48) n ( r + 1):P1A P1A (cid:48) ,∂ i ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = ˜ T L [ P (cid:48) ] [ P ] C3 P1A P1A (cid:48) ,∂ t ( S ) = ˜ T L [ P ] [ P (cid:48) ] ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] C3 P1B P1B (cid:48) , ∂ i ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = ˜ T L [ P (cid:48) ] [ P ] C3 P1B P1B (cid:48) . ∂ t ( S ) = ˜ T L [ P ] [ P (cid:48) ] ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] C3 For a type C5 or C6 saddle S : p n ( r ) ↔ p (cid:48) n ( r + 1):P2 P2 (cid:48) ,∂ i ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ i ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = ˜ T L [ P (cid:48) ] [ P ] C5,C6
P2 P2 (cid:48) .∂ t ( S ) = ˜ T L [ P ] [ P (cid:48) ] ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] ∂ t ( S ) = ˜ T L [ P ] [ P (cid:48) ] ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] C5,C6
At the head and tail of each arrow we indicate the type of the corresponding planar tangle. Below thehead (tail) of each arrow we indicate the corresponding map that could constitute the terminal (initial)factor of a residual term, where the first row corresponds to [ p ] ↔ [ p (cid:48) ] and the second row corresponds to[ p ] ↔ [ p (cid:48) ] . Saddles that do not yield initial or terminal factors are not included in the list.As an example, consider a type W2 saddle S : p (cid:48) n − ( r ) → p n +1 ( r ), as described by diagram (18). Firstconsider maps out of C P (cid:48) = C [ P (cid:48) ] ⊕ C [ P (cid:48) ] that could yield initial factors. When we reduce on P (cid:48) , weidentify C P (cid:48) with the subspace C P (cid:48) ⊗ x of C [ P (cid:48) ] = C P (cid:48) ⊗ A . So maps out of C [ P (cid:48) ] = C P (cid:48) do not yieldinitial factors, as we indicate by putting a zero in the lower-right corner of table (29), and we need onlyconsider maps out of C P (cid:48) ⊗ x . In fact we do have a nonzero map ˜ T L [ P ] [ P (cid:48) ] = P P P (cid:48) ( S ) ⊗ (cid:15) : C P (cid:48) ⊗ x → C P ,as we indicate by putting ∂ i ( S ) = ˜ T L [ P ] [ P (cid:48) ] in the upper-right corner of table (29). Next consider mapsinto C P = C [ P ] ⊕ C [ P ] that could yield terminal factors. When we reduce on P , we identify C P with thesubspace C P ⊗ A of C [ P ] = C P ⊗ A . So maps into C [ P ] = C P do not yield terminal factors, as we indicate HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 33 by putting a zero in the upper-left corner of table (29), and we need only consider maps into C P ⊗ A . Thereare no such maps that could yield a terminal factor, so we put a zero in the lower-left corner of table (29).The remaining tables in our list are constructed similarly.From the above list, we find that the terms of ∂ + T + that could yield residual terms are of the followingforms. We have pairs of type (W3,C2):P1A P1A (cid:48) P2 (cid:48)(cid:48) ,Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W3 C2
P2 P1B (cid:48)
P1B (cid:48)(cid:48) , ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C2 W3 (30) P2 P1A (cid:48)
P1A (cid:48)(cid:48) , ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C2 W3
P1B P1B (cid:48) P2 (cid:48)(cid:48) .Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W3 C2 (31)We have pairs of type (W3,C3):P1A P1A (cid:48)
P1A (cid:48)(cid:48) ,Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W3 C3
P1B P1B (cid:48)
P1B (cid:48)(cid:48) ,Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W3 C3 (32) P1A P1A (cid:48)
P1A (cid:48)(cid:48) , ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C3 W3
P1B P1B (cid:48)
P1B (cid:48)(cid:48) . ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C3 W3 (33)We have pairs of type (W4,C2):P1A P2 (cid:48) P2 (cid:48)(cid:48) , ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C2 W4
P2 P2 (cid:48)
P1B (cid:48)(cid:48) .Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W4 C2 (34)We have pairs of type (W4,C5):P2 P2 (cid:48) P2 (cid:48)(cid:48) ,Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W4 C5
P2 P2 (cid:48) P2 (cid:48)(cid:48) . ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C5 W4 (35)We have pairs of type (W4,C6):P2 P2 (cid:48) P2 (cid:48)(cid:48) ,Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] Q [ P (cid:48) ] [ P ] ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W4 C6
P2 P2 (cid:48) P2 (cid:48)(cid:48) . ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C6 W4 (36)We claim that pairs of saddles S : p → p (cid:48) and S : p (cid:48) → p (cid:48)(cid:48) of the forms shown in tables (30), (34), and(36) are always disjoint. It follows that the corresponding induced saddles [ p ] k → [ p (cid:48) ] k and [ p (cid:48) ] k → [ p (cid:48)(cid:48) ] k aredisjoint, and thus do not yield residual terms. We will show this for the first form in table (30). Let ( s (cid:48) , s (cid:48) )denote the saddle arcs of S in p (cid:48) , and let ( t (cid:48) , t (cid:48) ) denote the saddle arcs of S in p (cid:48) . Since S is of type W3and S is of type C2, we have s (cid:48) < s (cid:48) ≤ B (cid:48) < C (cid:48) , t (cid:48) ≤ B (cid:48) < C (cid:48) ≤ t (cid:48) . From Remark 4.5 and Lemma 7.2, it follows that s (cid:48) < s (cid:48) ≤ t (cid:48) ≤ B (cid:48) < C (cid:48) ≤ t (cid:48) , so the pair ( S , S ) is disjoint.We claim that pairs of saddles S : p → p (cid:48) and S : p (cid:48) → p (cid:48)(cid:48) of the forms shown in table (31) yield residualterms that are zero. We will show this for the first form in table (31). When we reduce on P (cid:48)(cid:48) , we identify C P (cid:48)(cid:48) with the subspace C P (cid:48)(cid:48) ⊗ x of C [ P (cid:48)(cid:48) ] = C P (cid:48)(cid:48) ⊗ A . From diagrams (23) and (19) for type C2 and W3saddles, we have ˜ T L [ P (cid:48) ] [ P ] = [ T RP (cid:48) P ( S )] ⊗ η, Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] = Q P (cid:48)(cid:48) P (cid:48) ( S ) ⊗ A . Since the image of Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] lies in C P (cid:48)(cid:48) ⊗ A , the residual term is zero.We claim that pairs of saddles S : p → p (cid:48) and S : p (cid:48) → p (cid:48)(cid:48) of the forms shown in tables (32), (33),and (35) yield residual terms of the form ˜ T LP (cid:48)(cid:48) P (cid:48) ( S ) Q P (cid:48) P ( S ) or Q P (cid:48)(cid:48) P (cid:48) ( S ) ˜ T LP (cid:48) P ( S ), both of which occurin ∂ + T + . We will show this for the first form in table (32). When we reduce on P , we identify C P with thesubspace C P ⊗ x of C [ P ] = C P ⊗ A . When we reduce on P (cid:48)(cid:48) , we identify C P (cid:48)(cid:48) with the subspace C P (cid:48)(cid:48) ⊗ x of C [ P (cid:48)(cid:48) ] = C P (cid:48)(cid:48) ⊗ A . From diagrams (19) and (24) for type W3 and C3 saddles, we have Q [ P (cid:48) ] [ P ] ( S ) = Q P (cid:48) P ( S ) ⊗ A , ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] ( S ) = ˜ T LP (cid:48)(cid:48) P (cid:48) ( S ) ⊗ A . So we obtain the residual term ˜ T LP (cid:48)(cid:48) P (cid:48) ( S ) Q P (cid:48) P ( S ) : C P → C P (cid:48)(cid:48) . (cid:3) Lemma 10.2.
Terms of ∂ + T + that correspond to pairs of saddles that are both of type (3) do not yieldreduction factors.Proof. Terms of ∂ T + that correspond to pairs of saddles that are both of type (3) take one of the followingforms:˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] Q [ P (cid:48) ] [ P ] , ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] Q [ P (cid:48) ] [ P ] , Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] , Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] , where, for example, Q [ P (cid:48) ] [ P ] and ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] are maps corresponding to type (3) saddles [ p ] → [ p (cid:48) ] and[ p (cid:48) ] → [ p (cid:48)(cid:48) ] that occur in the resolution of T , which are induced by type-changing saddles p → p (cid:48) and p (cid:48) → p (cid:48)(cid:48) that occur in the resolution of T . For tangle types P of type P1A, we could obtain reduction factors α ( P, S , S ) = α m ( S ) ∂ i ( S ) : C P (cid:48)(cid:48) → C P , β ( P, S , S ) = ∂ t ( S ) β m ( S ) : C P ⊗ A → C P (cid:48)(cid:48) , where ∂ i ( S ) : C P (cid:48)(cid:48) → C T + and α m ( S ) : C T + → C P are maps corresponding to saddles induced by type-changing saddles S : p (cid:48)(cid:48) → p (cid:48) and S : p (cid:48) → p , and β m ( S ) : C P ⊗ A → C T + and ∂ t ( S ) : C T + → C P (cid:48)(cid:48) aremaps corresponding to saddles induced by type-changing saddles S : p → p (cid:48) and S : p (cid:48) → p (cid:48)(cid:48) . For tangletypes P of type P1B, we could obtain reduction factors α ( P, S , S ) = α m ( S ) ∂ i ( S ) : C P (cid:48)(cid:48) → C P ⊗ x, β ( P, S , S ) = ∂ t ( S ) β m ( S ) : C P → C P (cid:48)(cid:48) , where ∂ i ( S ) : C P (cid:48)(cid:48) → C T + and α m ( S ) : C T + → C P ⊗ x are maps corresponding to saddles induced bytype-changing saddles S : p (cid:48)(cid:48) → p (cid:48) and S : p (cid:48) → p , and β m ( S ) : C P → C T + and ∂ t ( S ) : C T + → C P (cid:48)(cid:48) aremaps corresponding to saddles induced by type-changing saddles S : p → p (cid:48) and S : p (cid:48) → p (cid:48)(cid:48) .The maps ∂ i ( S ) and ∂ t ( S ) that appear in the reduction factors α ( P, S , S ) and β ( P, S , S ) are theinitial and terminal factors that we described in the proof of Lemma 10.1. We will refer to the maps α m ( S )and β m ( S ) as the middle factors of α ( P, S , S ) and β ( P, S , S ). For tangle types P of type P1A, we havemiddle factors α m ( S ) : C T + → C P , β m ( S ) : C P ⊗ A → C T + . For tangle types P of type P1B, we have middle factors α m ( S ) : C T + → C P ⊗ x, β m ( S ) : C P → C T + . Using the results of Sections 7 and 8, for each type-changing saddle S : p ↔ p (cid:48) that occurs in the resolutionof T we indicate the possible middle factors that could be obtained:For a type W2 saddle S : p n +1 ( r ) ↔ p (cid:48) n − ( r ):P1B P1A (cid:48) .β m ( S ) = ˜ T L [ P (cid:48) ] [ P ]
00 0 W2 (37) HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 35
For a type W3 saddle S : p n +1 ( r ) ↔ p (cid:48) n − ( r ):P1A P1A (cid:48) ,β m ( S ) = Q [ P (cid:48) ] [ P ] α m ( S ) = Q [ P (cid:48) ] [ P ] W3 P1B P1B (cid:48) .β m ( S ) = Q [ P (cid:48) ] [ P ] α m ( S ) = Q [ P (cid:48) ] [ P ] W3 For a type C1 saddle S : p n ( r ) ↔ p (cid:48) n ( r + 1):P1A P1A (cid:48) , α m ( S ) = ˜ T L [ P (cid:48) ] [ P ] C1 P1A P1A (cid:48) , α m ( S ) = ˜ T L [ P ] [ P (cid:48) ] C1 P1B P1B (cid:48) ,β m ( S ) = ˜ T L [ P (cid:48) ] [ P ]
00 0 C1 P1B P1B (cid:48) . β m ( S ) = ˜ T L [ P ] [ P (cid:48) ] C1 For a type C3 saddle S : p n ( r ) ↔ p (cid:48) n ( r + 1):P1A P1A (cid:48) ,β m ( S ) = ˜ T L [ P (cid:48) ] [ P ] α m ( S ) = ˜ T L [ P (cid:48) ] [ P ] C3 P1A P1A (cid:48) , β m ( S ) = ˜ T L [ P ] [ P (cid:48) ] α m ( S ) = ˜ T L [ P ] [ P (cid:48) ] C3 P1B P1B (cid:48) ,β m ( S ) = ˜ T L [ P (cid:48) ] [ P ] α m ( S ) = ˜ T L [ P (cid:48) ] [ P ] C3 P1B P1B (cid:48) . β m ( S ) = ˜ T L [ P ] [ P (cid:48) ] α m ( S ) = ˜ T L [ P ] [ P (cid:48) ] C3 At the head and tail of each arrow we indicate the type of the corresponding planar tangle. Below the head(tail) of each arrow we indicate the corresponding map that could contribute to a middle factor α m ( β m ),where the first row corresponds to [ p ] ↔ [ p (cid:48) ] and the second row corresponds to [ p ] ↔ [ p (cid:48) ] . Type-changingsaddles saddles S : p ↔ p (cid:48) that do not yield middle factors are not included in the list.As an example, consider the middle factors that can be obtained from a type W2 saddle S : p n +1 ( r ) → p (cid:48) n − ( r ), as described by diagram (18). Since p n +1 ( r ) is of type P1B and p n − ( r ) is of type P1A, it is notpossible to obtain a middle factor β m ( S ) mapping out of C P ⊗ A or α m ( S ) mapping into C P (cid:48) ⊗ A , so we putzeros in the lower-left and upper-right corners of table (37). There are potential middle factors of the form β m ( S ) : C P → C T + , α m ( S ) : C T + → C P (cid:48) . From diagram (18), we see that we have a nonzero map˜ T L [ P (cid:48) ] [ P ] = Q P (cid:48) P ( S ) ⊗ η : C P → C P (cid:48) ⊗ A, so we obtain the middle factor β m ( S ) = ˜ T L [ P (cid:48) ] [ P ] indicated in the upper-left corner of table (37). Diagram(18) shows there is no nonzero map into C P (cid:48) , so we set the lower-right corner of table (37) to zero. Theremaining tables in our list are constructed similarly.From the above list, we find that the terms of ∂ + T + that could yield reduction factors are of the followingforms. We have pairs of type (W2,W3):P1B P1A (cid:48) P1A (cid:48)(cid:48) .β m ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] W2 W3 (38)
We have pairs of type (W3,C2):P1A P1A (cid:48) P2 (cid:48)(cid:48) ,β m ( S ) = Q [ P (cid:48) ] [ P ] ∂ t ( S ) = ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W3 C2 (39) P2 (cid:48)(cid:48)
P1B (cid:48)
P1B .∂ i ( S ) = ˜ T L [ P (cid:48) ] [ P (cid:48)(cid:48) ] α m ( S ) = Q [ P ] [ P (cid:48) ] C2 W3 (40)We have pairs of type (W3,C3):P1A P1A (cid:48)
P1A (cid:48)(cid:48) ,β m ( S ) = Q [ P (cid:48) ] [ P ] ∂ t ( S ) = ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W3 C3
P1A P1A (cid:48)
P1A (cid:48)(cid:48) ,β m ( S ) = ˜ T L [ P (cid:48) ] [ P ] ∂ t ( S ) = Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] C3 W3
P1B (cid:48)(cid:48)
P1B (cid:48)
P1B ,∂ i ( S ) = ˜ T L [ P (cid:48) ] [ P (cid:48)(cid:48) ] α m ( S ) = Q [ P ] [ P (cid:48) ] C3 W3
P1B (cid:48)(cid:48)
P1B (cid:48)
P1B .∂ i ( S ) = Q [ P (cid:48) ] [ P ] α m ( S ) = ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] W3 C3
We claim that in each case the reduction factor is zero. First consider the form shown in table (38). Fromdiagrams (18) and (19) for type W2 and W3 saddles, we have that˜ T L [ P (cid:48) ] [ P ] = Q P (cid:48) P ( S ) ⊗ η : C P → C P (cid:48) ⊗ A, Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] = Q P (cid:48)(cid:48) P (cid:48) ( S ) ⊗ A : C P (cid:48) ⊗ A → C P (cid:48)(cid:48) ⊗ A. Since the image of Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] lies in C P (cid:48)(cid:48) ⊗ A , it follows that β ( P, S , S ) = Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] = 0.Next consider the form shown in table (39). From diagrams (19) and (23) for type W3 and C2 saddles, wehave that Q [ P (cid:48) ] [ P ] = Q P (cid:48) P ( S ) ⊗ A : C P ⊗ A → C P (cid:48) ⊗ A, ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] = [ T RP (cid:48)(cid:48) P (cid:48) ] ⊗ (cid:15) : C P (cid:48) ⊗ A → W (cid:48)(cid:48) . Since (cid:15) (1 A ) = 0, it follows that β ( P, S , S ) = ˜ T L [ P (cid:48)(cid:48) ] [ P (cid:48) ] Q [ P (cid:48) ] [ P ] = 0. Next consider the form shown intable (40). From diagrams (23) and (19) for type C2 and W3 saddles, we have that˜ T L [ P (cid:48) ] [ P (cid:48)(cid:48) ] = [ T RP (cid:48) P (cid:48)(cid:48) ] ⊗ η : W (cid:48)(cid:48) → C P (cid:48) ⊗ A, Q [ P ] [ P (cid:48) ] = Q P P (cid:48) ( S ) ⊗ A : C P (cid:48) ⊗ A → C P ⊗ A. Since the image of Q [ P ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P (cid:48)(cid:48) ] lies in C P ⊗ A , it follows that α ( P, S , S ) = Q [ P (cid:48)(cid:48) ] [ P (cid:48) ] ˜ T L [ P (cid:48) ] [ P ] = 0.The arguments for the remaining forms are similar. (cid:3) Lemma 10.3.
All the reduction terms obtained from ∂ T + occur in ∂ T + .Proof. Lemma 6.5 states that terms of ∂ T + corresponding to saddles of type (1) and (2) yield a reductionfactor β ( P ) = ˜ ∂ c P for each tangle type P of type P1A and a reduction factor α ( P ) = ˜ ∂ c P for each tangletype P of type P1B.We now consider the additional reduction factors that are obtained if we include saddles of type (3). ByLemma 10.2, we can ignore terms of ∂ + T + corresponding to pairs of saddles that are both of type (3). Type(3) saddles [ p ] → [ p (cid:48) ] and [ p ] → [ p (cid:48) ] obtained from the resolution of T are induced by a type-changingsaddle S : p → p (cid:48) obtained from the resolution of T . Using the results of Sections 7 and 8, for each type-changing saddle S : p → p (cid:48) obtained from the resolution of T we indicate the corresponding reduction factors: HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 37
For a type W2 saddle S : p n +1 ( r ) ↔ p (cid:48) n − ( r ):P1B P1A (cid:48) .β ( P, S ) = Q P (cid:48) P ( S ) α ( P (cid:48) , S ) = Q P (cid:48) P ( S ) W2 For a type C1 saddle S : p n ( r ) ↔ p (cid:48) n ( r + 1):P1A P1A (cid:48) ,β ( P, S ) = ˜ T LP (cid:48) P ( S ) + ˜ T RP (cid:48) P ( S ) 0 C1 P1A P1A (cid:48) , β ( P (cid:48) , S ) = ˜ T LP P (cid:48) ( S ) + ˜ T RP P (cid:48) ( S ) C1 P1B P1B (cid:48) , α ( P (cid:48) , S ) = ˜ T LP (cid:48) P ( S ) + ˜ T RP (cid:48) P ( S ) C1 P1B P1B (cid:48) .α ( P, S ) = ˜ T LP P (cid:48) ( S ) + ˜ T RP P (cid:48) ( S ) 0 C1 For a type C2 saddle S : p n ( r ) ↔ p (cid:48) n ( r + 1):P1A P2 (cid:48) ,β ( P, S ) = ˜ T LP (cid:48) P ( S ) + ˜ T RP (cid:48) P ( S ) 0 C2 P1B P2 (cid:48) .α ( P, S ) = ˜ T LP P (cid:48) ( S ) + ˜ T RP P (cid:48) ( S ) 0 C2 At the head and tail of each arrow we indicate the type of the corresponding planar tangle. Below the head(tail) of each arrow we indicate the reduction factor α ( β ) obtained via reduction on the correspondingtangle type. Saddles that do not yield reduction factors are not included in the list.We find that for each type W2 saddle S : p n +1 ( r ) → p (cid:48) n − ( r ) we obtain the reduction term˜ ∂ c P (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ c P = ˜ ∂ LP (cid:48) Q P (cid:48) P ( S ) + Q P (cid:48) P ( S ) ˜ ∂ LP , where the equality was shown in the proof of Lemma 9.4. We also obtain the following reduction terms fora pair of saddles ( S , S ) of type (W2,C1) and (W2,C2):˜ T LP (cid:48)(cid:48) P (cid:48) ( S ) Q P (cid:48) P ( S ) + ˜ T RP (cid:48)(cid:48) P (cid:48) ( S ) Q P (cid:48) P ( S ) , Q P (cid:48)(cid:48) P (cid:48) ( S ) ˜ T LP (cid:48) P ( S ) + Q P (cid:48)(cid:48) P (cid:48) ( S ) ˜ T RP (cid:48) P ( S ) . We can view S : p → p (cid:48) and S : p (cid:48) → p (cid:48)(cid:48) as two sides of a commuting square of saddles. Let S : p → p (cid:48)(cid:48) and S : p → p denote the sides opposite to S and S . If S and S constitute two sides of a commutingsquare of interleaved saddles, then˜ T RP (cid:48)(cid:48) P (cid:48) ( S ) Q P (cid:48) P ( S ) = ˜ T LP (cid:48)(cid:48) P ( S ) Q P P ( S ) , Q P (cid:48)(cid:48) P (cid:48) ( S ) ˜ T RP (cid:48) P ( S ) = Q P (cid:48)(cid:48) P ( S ) ˜ T LP P ( S ) . If S and S constitute two sides of a commuting square of nested saddles, then˜ T RP (cid:48)(cid:48) P (cid:48) ( S ) Q P (cid:48) P ( S ) = Q P (cid:48)(cid:48) P ( S ) ˜ T LP P ( S ) , Q P (cid:48)(cid:48) P (cid:48) ( S ) ˜ T RP (cid:48) P ( S ) = ˜ T LP (cid:48)(cid:48) P ( S ) Q P P ( S ) . If S and S constitute two sides of a commuting square of disjoint saddles, then the contributions of ( S , S )and ( S , S ) cancel. In each case, the residual terms we obtain occur in ∂ T + . (cid:3) Lemmas 10.1 and 10.3 give
Lemma 10.4.
All the residual and reduction terms of ∂ red occur in ∂ T + . Spectral sequence
In this section we use 1-tangle diagrams in the annulus to construct a spectral sequence that converges toreduced Khovanov homology. If we forget the bigrading of the chain complex ( C T ± , ∂ T ± ), we obtain a chaincomplex ( C T , ∂ ± T ), where ∂ T ± = ∂ T + ∂ ± T ; note in particular that C T = C T + = C T − as ungraded vectorspaces and ∂ T = ∂ T + = ∂ T − as maps of ungraded vector spaces. We define a Z -grading on C T by C T = (cid:77) s C sT , where C sT = (cid:77) { A c ( T i ) | i ∈ I such that r ( i ) = s } . We refer to the grading s as the resolution degree . We indicate that a homogeneous vector v ∈ C T hasresolution degree s by using a superscript: v ( s ) . The resolution degrees of the maps ∂ T and ∂ ± T that appear in the differential ∂ T ± = ∂ T + ∂ ± T are ( ∂ T ) (1) and ( ∂ ± T ) (2) . The fact that ( ∂ T ) = 0 holds in each resolutiondegree thus implies: Lemma 11.1.
We have ( ∂ T ) = ( ∂ T ± ) = 0 . We can define a filtration · · · ⊃ K − T ⊃ K T ⊃ K T ⊃ · · · of the chain complex C T by K rT = (cid:77) s ≥ r C sT . Using Theorem 3.1, we thus obtain Theorem 1.2 from the Introduction:
Theorem 11.2.
There is a spectral sequence with E page ( H ∂ T ( C T ) , d ± ) , where d ± ([ x ]) = [ ∂ ± T x ] , thatconverges to the reduced Khovanov homology of the link T ± . Example
We now illustrate Theorem 3.1 with an example. Consider the oriented disk 2-tangle diagram T D shownin Figure 10. Let T denote the corresponding oriented annular 1-tangle diagram, as shown in Figure 1. Weresolve the crossings of T to obtain the following planar tangles and saddles: p − (0) p (0) p − (1) .p − (0)For T + we have n + ( T + ) = 2 and n − ( T + ) = 1, so the chain complex ( C T + , ∂ T + ) is F [0 , F [0 , A [1 , , F [0 , ˙ η η ˙ η and we obtain the reduced Khovanov homology for the unknot:Khr( T + ) = F [0 , . For T − we have n + ( T − ) = 3 and n − ( T − ) = 0, so the chain complex ( C T − , ∂ T − ) is F [2 , F [0 , A [3 , , F [2 , ˙ η
00 ˙ η and we obtain the reduced Khovanov homology for the right trefoil:Khr( T − ) = F [0 , ⊕ F [2 , ⊕ F [3 , . The E page of the spectral sequence for T is H ∂ T ( C T ) = F ·{ [1] (0) , [(1 , (1) , [1 A ] (2) } . For T + , the differential is d +2 is given by d +2 ([1]) = [1 A ] , d +2 ([(1 , , d +2 ([1 A ]) = 0 . For T − , the differential is d − = 0. HOVANOV HOMOLOGY VIA 1-TANGLE DIAGRAMS IN THE ANNULUS 39
Figure 10.
Example disk 2-tangle diagram T D . Acknowledgments
The author would like to thank Matthew Hedden, Ciprian Manolescu, and Peter Ozsv´ath for helpfuldiscussions.
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