aa r X i v : . [ m a t h . G T ] M a y HOMFLYPT HOMOLOGY OVER Z DETECTS UNLINKS
HAO WU
Abstract.
We apply the Rasmussen spectral sequence to prove that the Z -graded vector space structureof the HOMFLYPT homology over Z detects unlinks. Our proof relies on a theorem of Batson and Seedstating that the Z -graded vector space structure of the Khovanov homology over Z detects unlinks. Introduction
If the Jones polynomial determines certain properties of links, then so does the HOMFLYPT polynomial.This is because the latter specializes to the former. A reasonable question is whether the same is true forcategorifications of these link polynomials. The answer is far less clear since there is no simple “specialization”from the HOMFLYPT homology to the Khovanov homology. Instead, we have the Rasmussen spectralsequence. In this manuscript, we study a particular problem of this type: unlink detection.Kronheimer and Mrowka proved in [7] that the vector space structure of the Khovanov homology over Z = Z / Z detects the unknot. Later, Hedden and Ni proved in [3] that the module structure of theKhovanov homology over Z detects unlinks. Improving these results, Batson and Seed proved in [1] thatthe Z -graded vector space structure of the Khovanov homology over Z detects unlinks.Upon closer inspection of the Rasmussen spectral sequence, we found that, if the HOMFLYPT homologyof a link is isomorphic to that of an unlink, then its Khovanov homology is also isomorphic to that of anunlink. In particular, the HOMFLYPT homology over Z detects unlinks. The following is our main result. Theorem 1.1.
Suppose that B is a closed braid of m components. Let R be the polynomial ring over Z generated by m variables, graded so that each variable is homogeneous of degree . Then the following twostatements are equivalent.(1) B is Markov equivalent to the m -component unlink.(2) As Z -graded vector spaces over Z , the unreduced HOMFLYPT homology H ( B ) of B over Z isgiven by H i,j ( B ) ∼ = ( R ⊕ ( ml ) { l } x if i = m − l, ≤ l ≤ m, and j = − m, otherwise,where H i,j ( B ) is the subspace of H ( B ) with horizontal grading i and vertical grading j , and { s } x means shifting the x -grading by s . The proof of Theorem 1.1 in Section 2 is itself quite short. But to understand this proof, one needs toknow some basic properties of the HOMFLYPT homology and the Rasmussen spectral sequence connectingit to the Khovanov homology. So, for the convenience of the reader, we include a brief review of thesesubjects in Appendix A. We need to point out that the normalization of H ( B ) in the current paper is givenin Appendix A, which is different from the ones in [5, 11].Also, there is a slight refinement of Theorem 1.1, in which one does not assume B has exactly m compo-nents to start with. This is because, according to Theorem 3.3 below, the Z -graded vector space structureof H ( B ) actually determines the number of components of B . We will state this refinement in Corollary 3.4below. Mathematics Subject Classification.
Primary 57M27.
Key words and phrases.
HOMFLYPT homology, Rasmussen spectral sequence, Khovanov homology, unknot, unlink.The author was partially supported by NSF grant DMS-1205879. . Detection of Unlinks
We prove Theorem 1.1 in this section. Let us establish some technical lemmas first.
Lemma 2.1.
Let B be a closed braid of m components with a marking, in which x i is the variable as-signed to a marked point on the i -th component for i = 1 , . . . , m . Then there is an obvious identification R B ∼ = Z [ x , . . . , x m ] . For i = 1 , . . . , m , there is a homogeneous R B -module homomorphism H ( B ) h i −→ H ( B ) that preserves the vertical grading, raises the horizontal grading by , lowers the x -grading by , and satisfiesthat d − ◦ h i + h i ◦ d − = x i · id H ( B ) .Proof. With out loss of generality, we assume there is a marked point on the i -th component B adjacent tothe marked point to which x i is assigned. That is, there is a segment A in this component connecting thesemarked points with no crossings or other marked points in its interior. Say y i is the variable assigned tothis adjacent marked points. Then the unnormalized HOMFLYPT chain complex of matrix factorizationsassociated to this segment A is 0 0ˆ C ( A ) = 0 / / Z [ x i , y i ] { } x o o x i + y i / / O O Z [ x i , y i ] x i + x i y i + y i o o / / O O . o o O O O O Define a homogeneous Z [ x i , y i ]-homomorphism h i : ˆ C ( A ) → ˆ C ( A ) by the diagram0 00 / / Z [ x i , y i ] { } x o o x i + y i / / O O Z [ x i , y i ] x i + x i y i + y i o o / / O O o o O O O O / / B B ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ Z [ x i , y i ] { } x o o @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) x i + y i / / O O Z [ x i , y i ] x i + x i y i + y i o o / / O O C C ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ . o o O O O O Clearly, h i satisfies the grading conditions in the lemma and commutes with both differentials d + and d v .Moreover,(2.1) d − ◦ h i + h i ◦ d − = ( x i + x i y i + y i ) · id ˆ C ( A ) . Denote by R the polynomial ring over Z generated by all the variables used in the marking of B . Recallthat, in Definition A.7, ˆ C ( B ) is defined by a big tensor product. Now we tensor h i with the identity maps ofall the other factors in that big tensor product. This extends h i to an R -linear endomorphism of double com-plex ( ˆ C ( B ) , d + , d v ) h i −→ ( ˆ C ( B ) , d + , d v ) satisfying the grading requirements in the lemma. After the necessarygrading shift, we get an R -linear endomorphism of double complex ( C ( B ) , d + , d v ) h i −→ ( C ( B ) , d + , d v ) satis-fying the grading requirements in the lemma. Recall that H ( B ) = H ( H ( C ( B ) , d + ) , d v ). Since h i commutes See Lemma A.9 below. ith both differentials of ( C ( B ) , d + , d v ), it induces a homogeneous R -module endomorphism H ( B ) h i −→ H ( B )satisfying the grading requirements in the lemma. Since R B is a quotient ring of R and the R -action on H ( B )factors through R B , H ( B ) h i −→ H ( B ) is also an R B -module homomorphism. Finally, by Lemma A.9, themultiplications of x i and y i are the same on H ( B ). Thus, by equation (2.1), d − ◦ h i + h i ◦ d − = x i · id H ( B ) . (cid:3) A key ingredient of the proof of Theorem 1.1 is the following algebraic lemma.
Lemma 2.2.
Let F be a field and R = F [ t , t , . . . , t m ] the polynomial ring with m variables over F gradedso that each t i is homogeneous of degree . Fix a positive integer N . For ≤ k ≤ m , denote by M k thegraded R -module M k := F [ t , t , . . . , t m ] / ( t Nk +1 , . . . , t Nm ) . Assume that the chain complex ( C ∗ , d ) = 0 → C m d m −−→ C m − d m − −−−→ · · · d −→ C d −→ C → satisfies:(1) C j is a graded R -module for j = 0 , , . . . , m ,(2) C j ∼ = M ⊕ ( mj ) k { jN } as graded F -spaces for j = 0 , , . . . , m , where { s } means shifting the modulegrading up by s ,(3) d j is a homogeneous R -module homomorphism of degree for j = 1 , . . . , m ,(4) for each i = 1 , . . . , k , there is an R -module homomorphism C ∗ h i −→ C ∗ +1 such that d ◦ h i + h i ◦ d = t Ni · id C ∗ . Then (2.2) H j ( C ∗ , d ) ∼ = 0 if j > m − k. In particulart, if k = m , then (2.3) H j ( C ∗ , d ) ∼ = ( M as graded F -spaces if j = 0 , if j ≥ , where M is the graded R -module M := F [ t , t , . . . , t m ] / ( t N , t N , . . . , t Nm ) .Proof. We prove Isomorphism (2.2) for 0 ≤ k ≤ m by inducting on k . If k = 0, it is obvious that H j ( C ∗ , d ) ∼ = 0if j > m since C j ∼ = 0 if j > m . So Isomorphism (2.2) is true if k = 0.Now assume Isomorphism (2.2) is true for k −
1. Assume ( C ∗ , d ) satisfies all the conditions in the lemma.Since d is an R -module homomorphism, t Nk · C ∗ is a subcomplex of C ∗ . Denote by D ∗ the quotient complex D ∗ := C ∗ /t Nk · C ∗ and by C ∗ π −→ D ∗ the standard quotient map. There is a short exact sequence(2.4) 0 → C ∗ t Nk −−→ C ∗ π −→ D ∗ → , which induces a long exact sequence(2.5) · · · ∆ j +1 −−−→ H j ( C ∗ ) t Nk −−→ H j ( C ∗ ) π −→ H j ( D ∗ ) ∆ j −−→ H j − ( C ∗ ) t Nk −−→ · · · A simple diagram chase shows that the connecting homomorphism ∆ j is a homogeneous R -module homo-morphism. By condition (3), the action by t Nk on H ( C ∗ ) is 0. So the long exact sequence (2.5) breaks intoshort exact sequences(2.6) 0 → H j ( C ∗ ) π −→ H j ( D ∗ ) ∆ j −−→ H j − ( C ∗ ) → , where both π and ∆ j are homogeneous R -homomorphisms. It is clear that D ∗ is a complex satisfying, for k −
1, all conditions in the lemma. So, by the induction hypothesis, H j ( D ∗ ) ∼ = 0 if j > m − k + 1. Thenshort exact sequences (2.6) implies that H j ( C ∗ ) ∼ = 0 if j > m − k . This completes the proof of Isomorphism(2.2).Now we focus on the case k = m and prove Isomorphism (2.3). By Isomorphism (2.2), we have that,if k = m , then H j ( C ∗ ) ∼ = 0 when j ≥
1. It remains to show that, when k = m , H ( C ∗ ) ∼ = M asgraded F -spaces. This comes down to a straightforward computation of the graded Euler characteristic.For any finitely generated graded R -module M , denote by gdim F M the graded dimension of M . That s, gdim F M = P α ∈ Z X α dim F M α , where M α is the homogeneous component of M of degree α . Since H j ( C ∗ ) ∼ = 0 if j ≥
1, the graded Euler characteristic of the chain complex C ∗ isgdim F H ( C ∗ ) = m X j =0 ( − j gdim F H j ( C ∗ ) = m X j =0 ( − j gdim F C j = m X j =0 ( − j X jN (cid:18) mj (cid:19) gdim F M m . But M m ∼ = R and gdim F R = − X ) m . Sogdim F H ( C ∗ ) = 1(1 − X ) m m X j =0 ( − j X jN (cid:18) mj (cid:19) = (1 − X N ) m (1 − X ) m = gdim F M . This shows that H ( C ∗ ) ∼ = M as graded F -spaces. (cid:3) In the proof of Theorem 1.1, we only need the following special case of Lemma 2.2.
Corollary 2.3.
Let R = Z [ x , x , . . . , x m ] the polynomial ring with m variables over Z graded so that each x i is homogeneous of degree . Assume that the chain complex ( C ∗ , d ) is of the form ← C − m d − m +2 ←−−−− C − m +2 d − m +4 ←−−−− · · · d m − l − ←−−−−− C m − l − d m − l ←−−−− C m − l d m − l +2 ←−−−−− · · · d m − ←−−− C m − d m ←−− C m ← , which satisfies:(1) C m − l ∼ = R ⊕ ( ml ) { l } x as graded R -modules for l = 0 , , . . . , m ,(2) d m − l is a homogeneous R -module homomorphism of degree for l = 0 , . . . , m − ,(3) for each i = 1 , . . . , m , there is an R -module homomorphism C ∗ h i −→ C ∗ +2 such that d ◦ h i + h i ◦ d = x i · id C ∗ . Then H m − l ( C ∗ , d ) ∼ = ( Z [ x , x , . . . , x m ] / ( x i , . . . , x m ) { m } x as graded Z -spaces if l = m, otherwise.Proof. This is the F = Z , N = 2, k = m case of Lemma 2.2 with an alternative grading convention. (cid:3) Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Fix a marking of B . Pick a marked point on each component of B . Assume x i is thevariable assigned to the chosen marked points on the i -th component. Then there is an obvious identification R B ∼ = Z [ x , . . . , x m ].By Theorem A.10 and Example A.12, we know that statement (1) implies statement (2).Now assume that statement (2) is true. Then, by Theorem A.11 and Lemma 2.1, ( E ( B ) , d ) ∼ = ( H ( B ) , d − )is a chain complex satisfying all conditions in Corollary 2.3. This implies that E i,j ( B ) ∼ = ( Z [ x , x , . . . , x m ] / ( x i , . . . , x m ) {− m } q as graded Z -spaces if i = j = − m, x -grading and the quantum grading. By Theorem A.11, d k strictlylowers the vertical grading for all k ≥
2. But E ( B ) is supported on a single vertical grading. This means d k = 0 for all k ≥
2. Thus, E i,j ∞ ( B ) ∼ = E i,j ( B ) ∼ = ( Z [ x , x , . . . , x m ] / ( x i , . . . , x m ) {− m } q as graded Z -spaces if i = j = − m, H i ( B ) ∼ = ( Z [ x , . . . , x m ] / ( x , . . . , x m ) {− m } q as graded Z -spaces if i = 0 , B is Markov equivalent to the m -component unlink. (cid:3) emark . Using Lemma 2.2, one can also prove that, if the HOMFLYPT homology over Q of a closedbraid B is isomorphic to that of the m -component unlink, then the sl ( N ) homology over Q of B is alsoisomorphic to that of the m -component unlink.3. A Slight Refinement of Theorem 1.1
Our refinement of Theorem 1.1 is based on an earlier result of the author in [12] that the HOMFLYPThomology determines the number of components of a link. More precisely, the degree of the Hilbert poly-nomial of the HOMFLYPT homology is the number of components of the link minus one. Since this resultis stated over Q in [12], for the convenience of the reader, we re-state it over Z and provide a sketch of itsproof.First, let us recall the Hilbert polynomial, which is a direct consequence of Hilbert’s Syzygy Theorem.We state this theorem below. Theorem 3.1 (Hilbert’s Syzygy Theorem) . Let F be a field and R the polynomial ring R = F [ x , . . . , x m ] graded so that each x i is homogeneous of degree . We say that the grading of a graded R -module is even ifthis module contains no non-zero homogeneous elements of odd degrees.Assume that M is a finitely generated graded R -module whose grading is even. Denote by M n thehomogeneous component of M of degree n . Then there is an exact sequence of graded R modules → F l → F l − → · · · → F → F → M → , in which, • l ≤ m , • each F j is a finitely generated free graded module over R whose grading is even, • each arrow is a homogeneous homomorphism of R -modules preserving the grading.As a standard consequence, there is a polynomial P ( T ) ∈ Q [ T ] of degree at most m − such that dim F M T = P ( T ) for T ≫ . This P ( T ) is called the Hilbert polynomial of M . For a proof of Theorem 3.1, see for example [9]. Note that, although the graded module structure of M isused in the construction of its Hilbert polynomial, this polynomial is completely determined by the graded F -space structure of M . Definition 3.2.
Let B be a closed braid and R B the polynomial ring over Z generated by components of B , graded so that each component of B is homogeneous of degree 2. From its definition in Appendix A, the x -grading of H ( B ) is even. For each horizontal grading i and vertical grading j , we define P B,i,j ( T ) to be theHilbert polynomial of the graded R B -module H i,j ( B ) with respect to its x -grading. The Hilbert polynomialof H ( B ) is P B ( T ) = P ( i,j ) ∈ Z × Z P B,i,j ( T ). Here, note that the right hand side is a finite sum since H ( B ) isfinitely generated over R B .The following theorem describes how the Hilbert polynomial of the HOMFLYPT homology determinesthe number of components of a link. Theorem 3.3. [12, Theorem 1.2]
Suppose that B is a closed braid of m components. Then the Hilbertpolynomial P B ( T ) of H ( B ) is of degree m − . Theorem 3.3 is proved in [12] over Q . Its proof over Z is essentially the same. Here we only include asketch of its proof over Z . For more details, see [12]. Sketch of proof of Theorem 3.3.
First of all, H ( B ) is a finitely generated graded R B -module with an even x -grading. So, according to Theorem 3.1, deg T P B ( T ) ≤ m − F B ( a, x ) = X ( i,j ) ∈ Z × Z ( − j a i x T dim Z ˆ H i,j, T ( B ) , ˆ Q B ( a, T ) = X ( i,j ) ∈ Z × Z ( − j a i ˆ P B,i,j ( T ) , where ˆ H i,j, T ( B ) is the homogeneous component of the unnormalized HOMFLYPT homology ˆ H ( B ) ofhorizontal grading i , vertical grading j and x -grading 2 T , • ˆ P B,i,j ( T ) is the Hilbert polynomial of ˆ H i,j ( B ) with respect to the x -grading. ✒■ B + ■ ✒ B − ✻ ✻ B Figure 1. ˆ F B ( a, x ) is the normalization of the HOMFLYPT polynomial satisfying(3.1) ˆ F B ( a, x ) is invariant under transverse Markov moves, xa − F B + ( a, x ) − x − a F B − ( a, x ) = ( x − − x ) F B ( a, x ) ,F B ′ ( a, x ) = − a − F B ( a, x ) ,F U ( a, x ) = a − x − x , where • transverse Markov moves are all Markov moves except the negative stabilization/destabilization, • B + , B − and B are closed braids identical outside the part shown in Figure 1, • B ′ is obtained from the closed braid B by a negative stabilization, • U is the closed braid with a single strand.From this, we know that ˆ Q B ( a, T ) satisfies the skein relation:(3.2) ˆ Q B ( a, x ) is invariant under transverse Markov moves, a − ˆ Q B + ( a, T ) − a ˆ Q B − ( a, T + 1) = ˆ Q B ( a, T + 1) − ˆ Q B ( a, T ) , ˆ Q B ′ ( α, T ) = − a − ˆ Q B ( a, T ) , ˆ Q U ( a, T ) = 1 + a − . Using skein relation (3.2), one can apply the “computation tree argument” in [2] to ˆ Q ( a, T ) to establishthat deg T ˆ Q ( a, T ) ≥ m −
1. See [12, Section 4] for more details. This implies that deg T ˆ P B,i,j ( T ) ≥ m − i, j ) ∈ Z × Z . But H ( B ) is obtained from ˆ H ( B ) by shifting the horizontal and vertical gradings.So { ˆ P B,i,j ( T ) | ( i, j ) ∈ Z × Z } and { P B,i,j ( T ) | ( i, j ) ∈ Z × Z } are the same collection of polynomials of T .So deg T P B,i,j ( T ) ≥ m − i, j ) ∈ Z × Z . Since the leading coefficient of the Hilbert polynomial ofa graded module must be positive, this implies thatdeg T P B ( T ) = deg T X ( i,j ) ∈ Z × Z P B,i,j ( T ) = max { deg T P B,i,j ( T ) | ( i, j ) ∈ Z × Z } ≥ m − . Altogether we have deg T P B ( T ) = m − (cid:3) Using Theorem 3.3, we get a slightly stronger version of the detection theorem of unlinks. Note that,unlike in Theorem 1.1, we do not assume B has exactly m components in Corollary 3.4. Corollary 3.4.
Suppose that B is a closed braid. Let R be the polynomial ring over Z generated by m variables, graded so that each variable is homogeneous of degree . Then the following two statements areequivalent.(1) B is Markov equivalent to the m -component unlink.(2) As graded Z -spaces, the unreduced HOMFLYPT homology H ( B ) of B is given by H i,j ( B ) ∼ = ( R ⊕ ( ml ) { l } x if i = m − l, ≤ l ≤ m, and j = − m, otherwise, here H i,j ( B ) is the Z -subspace of H ( B ) with horizontal grading i and vertical grading j , and { s } x means shifting the x -grading by s .Proof. By Theorem 1.1, statement (1) implies statement (2). Now assume statement (2) is true. The Hilbertpolynomial of R is (cid:0) T + m − m − (cid:1) , which is of degree m − T . From this, one can see that deg T P B ( T ) = m − B has exactly m components. Thus, by Theorem 1.1, statement (1) is true. (cid:3) References [1] J. Batson, C. Seed,
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J. Knot Theory Ramifications, (2018), no. 1, 1850007(20 pages) Appendix A. HOMFLYPT Homology, Khovanov Homology and Rasmussen SpectralSequence over Z In this appendix, we briefly review the unreduced HOMFLYPT homology over Z and the Rasmussenspectral sequence over Z connecting the HOMFLYPT homology to the Khovanov homology. We use agrading convention that differs from the one in [11]. We would like to emphasize that, although the statementof Theorem 1.1 does not mention the module structure of H ( B ), its proof does use this structure. So we willinclude the definition of this module structure in Lemma A.9 and Theorem A.10 below.A.1. Matrix factorizations.
In [11], both the HOMFLYPT homology and the Khovanov homology aredefined using matrix factorizations with an integer homological grading.
Definition A.1.
Let R be a graded commutative ring with 1. In this manuscript, we call the grading of R and the grading of any graded R -module the x -grading . Let w be a homogeneous element of R of x -grading N . A graded matrix factorization of w over R is a diagram0 d l − / / M ld l − o o d l + / / M l +2 d l +2 − o o d l +2+ / / · · · d l +2 k +4 − o o d l +2 k − / / M l +2 k − d l +2 k − − o o d l +2 k − / / M l +2 kd l +2 k − o o d l +2 k + / / , d l +2 k +2 − o o where • each M l +2 j is a graded free R -module, • each d l +2 j + is a homogeneous R -module homomorphism preserving the x -grading, • each d l +2 j − is a homogeneous R -module homomorphism raising the x -grading by N , • d l +2 j +2+ ◦ d l +2 j + = 0, d l +2 j − − ◦ d l +2 j − = 0 and d l +2 j − ◦ d l +2 j − + d l +2 j +2 − ◦ d l +2 j + = w · id M l +2 j for j = 0 , , . . . , k , • the horizontal grading of this matrix factorization is defined so that M l +2 j has horizontal grading l + 2 j for j = 0 , , . . . , k .We call d + and d − the positive and negative differentials of the matrix factorization.In the definitions below, we will mostly use a special type of matrix factorizations – Koszul matrixfactorizations. efinition A.2. Let a and b be homogeneous elements of R . Then the graded Koszul matrix factorization K R ( a, b ) over R of ab is the following diagram K R ( r ) := 0 / / R { deg b } x | {z } − o o b / / R |{z} a o o / / , o o where a and b act by multiplication, the under-braces indicate the horizontal grading, and { s } x meansshifting the x -grading up by s .More generally, assume that a , b , . . . , a k , b k are homogeneous elements of R such that a b , . . . , a k b k areof the same x -grading. Then the graded Koszul matrix factorization K R a b · · · · · · a k b k over R of P kj =1 a j b j is the tensor product over R of the graded Koszul matrix factorizations of all rows ( a j , b j ). That is, K R a b · · · · · · a k b k := K R ( a , b ) ⊗ R K R ( a , b ) ⊗ R · · · ⊗ R K R ( a k , b k ) . Note that permuting the sequence { ( a , b ) , . . . , ( a k , b k ) } permutes the factors in the above tensor productand, therefore, does not change the isomorphism type of the Koszul matrix factorization K R a b · · · · · · a k b k .A.2. Matrix factorizations of MOY graphs.
Next, we recall the matrix factorizations of MOY graphsused to defined both the HOMFLYPT homology and the Khovanov homology. ✒ ✒■■
Figure 2.
Definition A.3.
An MOY graph Γ is an embedding of a directed graph in the plane so that • each vertex of Γ has valence 4 or 1, • each 4-valent vertex of Γ looks like the one in Figure 2.We call a 1-valent vertex of Γ an endpoint, and a 4-valent vertex an interior vertex.A marking of an MOY graph Γ consists of • a finite collection of marked points on Γ such that – none of the interior vertices are marked, – all endpoints are marked, – each edge of Γ contains at least one marked points, • an assignment to each marked point a distinct variable of degree 2. ✲ x i x s Γ i ; s ❃⑦ ❃⑦ x j x i x t x s Γ i,j ; s,t Figure 3.
Fix an MOY graph Γ and a marking of Γ. Assume that x , . . . , x n are all the variables assigned to markedpoints in Γ. Let R = Z [ x , . . . , x n ] graded so that deg x = · · · = deg x n = 2. Cut Γ at all of its marked oints. We get a collection of pieces Γ , . . . , Γ m , each of which is of one of the two types in Figure 3. Wedefine their matrix factorizations by the following. • If Γ q = Γ i ; s in Figure 3, then R q = Z [ x i , x s ] and ˆ C (Γ q ) = K R q ( x s + x s x i + x i , x s + x i ). • If Γ q = Γ i,j ; s,t in Figure 3, then R q = Z [ x i , x j , x s , x t ] andˆ C (Γ q ) = K R q (cid:18) x s + x t + x i + x j + x t ( x s + x i + x j ) x s + x t + x i + x j x i + x j ( x s + x i )( x s + x j ) (cid:19) {− } x . Definition A.4.
The matrix factorization of Γ is defined to beˆ C (Γ) := m O q =1 ( ˆ C (Γ q ) ⊗ R q R ) , where the big tensor product “ N mq =1 ” is taken over the ring R = Z [ x , . . . , x n ].Note that ˆ C (Γ) is a graded matrix factorization over R . It has two gradings: the horizontal grading andthe x -grading. Its positive differential d + raises the horizontal grading by 2 and preserves the x -grading. Itsnegative differential d − lowers the horizontal gradings by 2 and raises the x -grading by 6.A.3. The unnormalized HOMFLYPT complex of a tangle.Definition A.5.
For a tangle T , an arc of T is a part of T that starts and ends at crossings or endpointsand contains no crossings or endpoints in its interior. A marking of T consists of • a finite collection of marked points on T such that – all endpoints of T are marked, – none of the crossings of T is marked, – each arc of T contains at least one marked point, • an assignment that assigns to each marked point a distinct variable of degree 2.Let T be a tangle with a marking. Assume x , . . . , x n are all the variables assigned to marked points in T . The ring R := Z [ x , . . . , x n ] is graded so that deg x j = 2 for j = 1 , . . . , n .Cut T at all of its marked points. This cuts T into a collection { T , . . . , T m } of simple tangles, each ofwhich is of one of the three types in Figure 4 and is marked only at its endpoints. ✻ x s x i A ✒■ x s x i x t x j C + ■ ✒ x s x i x t x j C − Figure 4. If T q = A , then R q = Z [ x i , x s ] and the unnormalized HOMFLYPT complex ˆ C ( T q ) of T q is(A.1) 0 |{z} ˆ C ( A ) | {z } O O |{z} − O O , where ˆ C ( A ) is the matrix factorization defined in the previous subsection, and the under-braces indicate the vertical gradings .To define the complexes of crossings, we need the χ -maps. ✻ x s x i x t x j Γ χ / / ✒■■ ✒ x s x i x t x j Γ χ o o Figure 5.
Lemma A.6. [4, 5, 11]
Let Γ and Γ be the MOY graphs marked as in Figure 5, and R = Z [ x s , x t , x i , x j ] .Then there exist homogeneous morphisms of matrix factorizations ˆ C (Γ ) χ −→ ˆ C (Γ ) and ˆ C (Γ ) χ −→ ˆ C (Γ ) of x -degree satisfying χ ◦ χ ≃ ( x s − x j ) · id ˆ C (Γ ) and χ ◦ χ ≃ ( x s − x j ) · id ˆ C (Γ ) . Here, a map between twomatrix factorizations over R is a “morphism of matrix factorizations” if it is an R -module homomorphismpreserving the horizontal grading and commuting with both the positive and the negative differentials. ✒■ x s x i x t x j C + z z ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ $ $ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ✻ ✻ x s x i x t x j Γ ✒■■ ✒ x s x i x t x j Γ ■ ✒ x s x i x t x j C − d d ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ : : ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ Figure 6. ow consider the resolutions in Figure 6. If T q = C ± , then R q = Z [ x i , x j , x s , x t ] and the unnormalizedHOMFLYPT chain complex ˆ C ( T q ) of T q isˆ C ( C + ) or ˆ C ( C − )0 |{z} |{z} ˆ C (Γ ) { } h {− } x | {z } O O ˆ C (Γ ) {− } h { } x | {z } O O ˆ C (Γ ) { } h {− } x | {z } − χ O O ˆ C (Γ ) {− } h { } x | {z } χ O O |{z} − O O |{z} − O O , where the under-braces indicate the vertical gradings , { s } h means shifting the horizontal grading by s ,and { t } x means shifting the x -grading by t . Definition A.7.
We define the unnormalized HOMFLYPT chain complex ˆ C ( T ) associated to T to beˆ C ( T )) := m O q =1 ( ˆ C ( T q ) ⊗ R q R ) , where the big tensor product “ N mq =1 ” is taken over R = Z [ x , . . . , x n ].Note that ˆ C ( T ) has three gradings: the vertical grading, the horizontal grading and the x -grading. It hasthree differentials: • the vertical differential d v , which raises the vertical grading by 2 and preserves the two other gradings, • the positive differential d + , which raises the horizontal grading by 2 and preserves the two othergradings, • the negative differential d − , which preserves the vertical grading, lowers the horizontal grading by 2and raises the x -grading by 6.Moreover, these three differentials commute with each other. A.4.
HOMFLYPT homology.
The HOMFLYPT homology of links are defined using closed braid dia-grams of links. Let B be a closed braid with a marking. Assume B has b strands and its writhe is w . Recallthat the self linking number of B is sl ( B ) = w − b . Definition A.8.
The unnormalized unreduced HOMFLYPT homology of B is ˆ H ( B ) := H ( H ( ˆ C ( B ) , d + ) , d v ).The normalized HOMFLYPT chain complex of B is C ( B ) := ˆ C ( B ) { w − b } v { b − w } h , where { r } v { s } h means shifting the vertical grading by r and the horizontal grading by s . The unreduced HOMFLYPThomology of B is H ( B ) := H ( H ( C ( B ) , d + ) , d v ) = ˆ H ( B ) { w − b } v { b − w } h .Let B be a closed braid with a marking, and x , . . . , x n all the variables assigned to marked points on B .The ring R = Z [ x , . . . , x n ] is graded so that each x j is homogeneous of degree 2. From its definition, C ( B )is a Z -graded R -module, and all of its three differentials are homogeneous R -module homomorphisms. So H ( B ) := H ( H ( C ( B ) , d + ) , d v ) is a Z -graded R -module. Since we are working over Z , there is no difference between commuting and anti-commuting. emma A.9. [10, Lemma 3.4] If x i and x j are assigned to marked points on the same component of B ,then the multiplications by x i and by x j are the same on H ( B ) .In particular, this means that H ( B ) is a finitely generated Z -graded module over the graded ring R B := R/ ( { x i − x j | x i and x j are assigned to marked points on the same component of B } ) . The proof in [10] of this lemma is for sl ( N ) homology over Q . But that proof can be adapted to work for H ( B ) without essential changes. We leave details to the reader.Note that R B can be viewed as the polynomial ring over Z generated by components of B , with eachcomponent homogeneous of degree 2. Theorem A.10. [5, 6]
Suppose that B is a closed braid and R B is the polynomial ring over Z generatedby components of B , graded so that each component of B is homogeneous of degree . Then the unreducedHOMFLYPT homology H ( B ) of B over Z is a finitely generated Z -graded R B -module. Up to isomorphismsof Z -graded R B -modules, H ( B ) is independent of the marking and invariant under Markov moves. Theorem A.10 is proved over Q in [5] and over Z in [6]. These proofs can be adapted to the base field Z without essential changes. The R B -module structure of the HOMFLYPT homology is not explicitlyaddressed in [5, 6]. But the proofs of invariance in these papers are local. For each local marking changeor braid-like Reidemeister move, the isomorphism constructed in [5, 6] commutes with multiplications byvariables assigned to marked points that are outside the interior of the part of B affected by this localmarking change or braid-like Reidemeister move. By the definition of markings of closed braids, there isalways such a marked point on each component of B . This implies that the isomorphisms constructed in[5, 6] are isomorphisms of R B -modules. Again, we leave details to the reader.A.5. Rasmussen spectral sequence.
In [11], Rasmussen proved that the spectral sequence associated tothe column filtration of the double complex ( H ( C ( B ) , d + ) , d − , d v ) converges to the sl (2) Khovanov homology.Of course, the proof in [11] is over Q . But, again, this proof adapts to the base field Z without essentialchanges. Theorem A.11. [11]
Suppose that B is a closed braid and R B is the polynomial ring over Z generated bycomponents of B , graded so that each component of B is homogeneous of degree . Denote by H ( B ) theKhovanov homology of B over Z . Then, the spectral sequence { E r ( B ) } associated to the column filtrationof the double complex ( H ( C ( B ) , d + ) , d − , d v ) satisfies:(1) { E r ( B ) } is a spectral sequence of R B -modules.(2) Each page of { E r ( B ) } inherits the horizontal and the vertical gradings of C ( B ) . It also inherits the sl (2) quantum grading of C ( B ) given by the degree function deg x +3 deg h , where deg x and deg h arethe degree functions of the x - and the horizontal gradings.(3) E ( B ) ∼ = H ( B ) as R B -modules, where the isomorphism preserves the horizontal, vertical and sl (2) quantum gradings.(4) E ∞ ( B ) ∼ = H ( B ) as R B -modules, where the quantum grading of H ( B ) coincide with the sl (2) quantum grading of E ∞ ( B ) , and the homological grading of H ( B ) coincides with the grading on E ∞ ( B ) given by the degree function deg v − deg h , where deg v and deg h are the degree functions ofthe vertical and the horizontal gradings of E ∞ ( B ) .(5) Under the identification E ( B ) ∼ = H ( B ) , the differential d of the page E ( B ) is the differential d − of H ( B ) , which preserves the vertical and the sl (2) quantum gradings, and lowers the horizontalgrading by . For all k ≥ , the differential d k of the pages E k ( B ) strictly lowers the vertical grading. Recently, Naisse and Vaz proved in [8] that the Rasmussen spectral sequence collapses at its E -page. Forour purpose though, we do not need this general result. This is because that, if the HOMFLYPT homologyof a link is isomorphic to that of an unlink, then its Rasmussen spectral sequence collapses at its E -pagefor simple grading reasons. Example A.12.
Let U ⊔ m be the m -strand closed braid with no crossings. This is of course a diagram of the m -component unlink. We mark U ⊔ m by putting a single marked points on each component of U ⊔ m . Denote y x j the variable assigned to the marked point on the j -th component. Then R U ⊔ m = Z [ x , . . . , x m ] and C ( U ⊔ m ) = K R U ⊔ m x · · · · · · x m {− m } v { m } h .d v and d + both vanish for this complex. So H ( U ⊔ m ) ∼ = C ( U ⊔ m ) as Z -graded R B -modules. In other words,as Z -graded R B -modules, H i,j ( U ⊔ m ) ∼ = ( R ⊕ ( ml ) U ⊔ m { l } x if i = m − l, ≤ l ≤ m, and j = − m, H i,j ( U ⊔ m ) is the R U ⊔ m -submodule of H ( U ⊔ m ) with horizontal grading i and vertical grading j .The complex ( H ( U ⊔ m ) , d − ) is the Koszul complex (with non-standard gradings) over R U ⊔ m = Z [ x , . . . , x m ]for the sequence { x , . . . , x m } , which is regular in R U ⊔ m . So E i,j ( U ⊔ m ) ∼ = H i,j ( H ( U ⊔ m ) , d − ) ∼ = ( Z [ x , . . . , x m ] / ( x , . . . , x m ) { m } x if i = j = − m, E i,j ( U ⊔ m ) is the R U ⊔ m -submodule of E ( U ⊔ m ) with horizontal grading i and vertical grading j . Since d k strictly lowers the vertical grading for k ≥
2, we have that d k = 0 for k ≥
2. So E ∞ ( U ⊔ m ) ∼ = E ( U ⊔ m ).Thus, H i ( U ⊔ m ) ∼ = ( Z [ x , . . . , x m ] / ( x , . . . , x m ) {− m } q if i = 0 , H i ( U ⊔ m ) is the R U ⊔ m -submodule of H ( U ⊔ m ) with homological grading i , and {− m } q means shiftingthe quantum grading down by m . Here, note the difference between the x -grading and the sl (2) quantumgrading.In particular, when m = 1, U = U ⊔ is the unknot. And H i,j ( U ) ∼ = Z [ x ] if i = 1 , j = − , Z [ x ] { } x if i = − , j = − , H i ( U ) ∼ = ( Z [ x ] / ( x ) {− } q if i = 0 , Department of Mathematics, The George Washington University, Phillips Hall, Room 739, 801 22nd StreetNW, Washington DC 20052, USA. Telephone: 1-202-994-0653, Fax: 1-202-994-6760
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