Integral HOMFLY-PT and sl(n)-link homology

Using the diagrammatic calculus for Soergel bimodules, developed by B. Elias and M. Khovanov, as well as Rasmussen's spectral sequence, we construct an integral version of HOMFLY-PT and sl(n)-link homology.


Introduction
During the past half-decade categorification and, in particular, that of topological invariants has flourished into a subject of its own right. It has been a study finding connections and ramifications over a vast spectrum of mathematics, including areas such as low-dimensional topology, representation theory, algebraic geometry, as well as others. Following the original work of M. Khovanov on the categorification of the Jones polynomial in [8], came a spew of link homology theories lifting other quantum invariants. With a construction that utilized a tool previously developed in an algebra-geometric context, matrix factorizations, M. Khovanov and L. Rozansky produced the sl(n) and HOMFLY-PT link homology theories. Albeit computationally intensive, it was clear from the onset that thick interlacing structure was hidden within. The most insightful and influential work in uncovering these innerconnections was that of J. Rasmussen in [17], where he constructed a spectral sequence from the HOMFLY-PT to the sl(n)-link homology. This was a major step in deconstructing the pallet of how these theories come together, yet many structural questions remained and still remain unanswered, waiting for a new approach. Close to the time of the original work, M. Khovanov produced an equivalent categorification of the HOMFLY-PT polynomial in [9], but this time using Hochschild homology of Soergel bimodules and Rouquier complexes of [18]. The latter proved to be more computation-friendly and was used by B. Webster to calculate many examples in [24].
In the meantime, a new flavor of categorification came into light. With the work of A. Lauda and M. Khovanov on the categorification of quantum groups in [13], a diagrammatic calculus originating in the study of 2-categories arrived into the foreground. This graphical approach proved quite fruitful and was soon used by B. Elias and M. Khovanov to rewrite the work of Soergel in [4], and en suite by B. Elias and the author to repackage Rouquier's complexes and to prove that they are functorial over braid-cobordisms [5] (not just projectively functorial as was known before). An immediate advantage to this construction was the inherent ease of calculation, at least comparative ease, and the fact that it worked equally well over the integers as well as over fields.
As there has yet to be seen an integral version of either HOMFLY-PT or sl(n)-link homology, with the original Khovanov homology being defined over Z and torsion playing an interesting role, a natural question arose as to whether this graphical calculus could be used to define these. The definition of such integral theories is precicely the purpose of this paper. The one immediate disadvantage to the graphical approach is that at the present moment there does not exist a diagrammatic calculus for the Hochschild homology of Soergel bimodules. Hence, to define integral HOMFLY-PT homology, our paper takes a rather roundabout way, jumping between matrix factorizations and diagrammatic Rouquier complexes whenever one is deemed more advantageous than the other. For the sl(n) version of the story, we add the Rasmussen spectral sequence into the mix and essentially repeat his construction in our context. The organization of the paper is the following: in section 2 we give a brief account of the necessary tools (matrix factorizations, Soergel bimodules, Hochschild homology, Rouquier complexes, and corresponding diagrammatics) -the emphasis here is brevity and we refer the reader to more original sources for particulars and details; in sections 3 and 4 we describe the intergal HOMFLY-PT complex and prove the Reidemeister moves, utilizing all of the background in 2; section 5 is devoted to the Rasmussen spectral sequence and integral sl(n)-link homology, and we conclude it with some remarks and questions.
Throughout the paper we will refer to a positive crossing as the one labelled D + and negative crossing as the one labelled D − in figure 1. For resolutions of a crossing we will refer to D o and D s of figure 1 as the "oriented" and "singular" resolutions, respectively. We will use the following conventions for the HOMLFY-PT polynomial with P of the unknot being 1. Substuting a = q n we arrive at the quantum sl(n)-link polynomial.

Figure 1: Crossings and resolutions
Acknowledgments: I would like to thank my advisor M. Khovanov for all his time and help in exploring the subjects at hand over the past few years. In addition, I thank B. Elias and P. Vaz for helpful conversations and e-mail exchanges.

The toolkit
We will require some knowledge of matrix factorizations, Soergel bimodules and Rouquier complexes, as well as the corresponding diagrammatic calculus of Elias and Khovanov [4]. In this section the reader will find a brief survery of the necessary tools, and for more details we refer him to the following papers: for matrix factorizations [11], [17], for Soergel bimodules and Rouquier complexes and diagrammatics [4], [5], [9], [18], and for Hochschild homology [6], [9].

Matrix factorizations
Definition 1. Let R be a Noetherian commutative ring, w ∈ R, and C * , * ∈ Z, a free graded R-module. A Z-graded matrix factorization with potential w consists of C * and a pair of differentials d ± : A morphism of two matrix factorizations C * and D * is a homomorphism of graded Rmodules f : C * → D * that commutes with both d + and d − . The tensor product C * ⊗ D * is taken as the regular tensor product of complexes, and is itself a matrix factorization with diffentials d + and d − . A useful and easy exercise is the following: Lemma 2. Given two matrix factorizations C * and D * with potenials w c and w d , respectively, the tensor product C * ⊗ D * is a matrix factorization with potential w c + w d .
Remark. Following Rasmussen [17], we work with Z-graded, rather than Z/2Z-graded, matrix factorizations as in [11]. The Z-grading implies that (d In the case that w = 0, we acquire a new Z/2Z-graded chain complex structure with differential d + + d − . Supressing the underlying ring R and potential w, we will denote the category of graded matrix factorizations by mf . We also need the notion of complexes of matrix factorizations. If we visualize a collection of matrix factorizations as sitting horizontally in the plane at each integer level, with differentials d + and d − running right and left, respectively, we can think of morphisms {d v } between these as running in the vertical direction. If d 2 v = 0 we get a complex, i.e. all together we have that where we think of i as the horizontal grading and j as the vertical grading, and will denote these as gr h and gr v , respectively. In addition we will be taking tensor products of complexes of matrix factorizations (in the obvious way) and, just to add to the confusion we will also have homotopies of these complexes as well homotopies of matrix factorizations themselves. These notions will land us in different categories to which we now give some notation.
• hmf will denote the homotopy category of matrix factorizations

Diagrammatics of Soergel bimodules
The category of Soergel bimodules SC 1 is generated monoidally over R by objects B i , i ∈ I, which satisfy (Technically speaking this should be called the category of Bott-Samuelson bimodules and the "real" category of Soergel bimodules is gotten as described at the end of this section. See also [4] and [5] for more details.) The Grothendieck group of SC(I) is isomorphic to the Hecke algebra H of type A ∞ over the ring Z[t, t −1 ], with the class of B i being sent to a generator b i of H, and the class of R{1} being sent to t.
More concretely, the Soergel bimodule B i = R⊗ R i R{−1}, where R is a graded polynomial ring, {m} denotes the grading shift by m, and R i is the subring of invariants corresponding to the permutation (i, i + 1) under the natural action of S n on the variables. There is some flexibility as to the exact description of R, but in our case it will mainly be the ring Z[x 1 − x 2 , . . . , x n−1 − x n ] with deg x i = 2 (note that our grading shift of −1 in the definition of B i is absent from the contruction of [9]). We have that B ∅ = R itself, and One useful feature of this categorification is that it is easy to calculate the dimension of Hom spaces in each degree. Let HOM(M, N) def = m∈Z Hom(M, N{m}) be the graded vector space (actually an R-bimodule) generated by homogeneous morphisms of all degrees.
is a free left R-module, and its graded rank over R is given by (b i , b j ). For more information on this categorification and related topics we refer the reader to [4], and [19].
The graphical counterpart, which we will also refer to as SC 1 was given a diagrammatic presentation by generators and relations, allowing morphisms to be viewed as isotopy classes of certain graphs.
An object in SC 1 is given by a sequence of indices i , which is visualized as d points on the real line R, labelled or "colored" by the indices in order from left to right. Sometimes these objects are also called B i . Morphisms are given by pictures embedded in the strip R × [0, 1] (modulo certain relations), constructed by gluing the following generators horizontally and vertically: For instance, if "blue" corresponds to the index i and "red" to j, then the lower right generator is a morphism from jij to iji. The generating pictures above may exist in various colors, although there are some restrictions based on adjacency conditions. We can view a morphism as an embedding of a planar graph, satisfying the following properties: 1. Edges of the graph are colored by indices from 1 to n.
2. Edges may run into the boundary R × {0, 1}, yielding two sequences of colored points on R, the top boundary i and the bottom boundary j . In this case, the graph is viewed as a morphism from j to i .
3. Only four types of vertices exist in this graph: univalent vertices or "dots", trivalent vertices with all three adjoining edges of the same color, 4-valent vertices whose adjoining edges alternate in colors between i and j distant, and 6-valent vertices whose adjoining edges alternate between i and j adjacent.
The degree of a graph is +1 for each dot and -1 for each trivalent vertex. 4-valent and 6-valent vertices are of degree 0. The term graph henceforth refers to such a graph embedding.
By convention, we color the edges with different colors, but do not specify which colors match up with which i ∈ I. This is legitimate, as only the various adjacency relations between colors are relevant for any relations or calculations. We will specify adjacency for all pictures, although one can generally deduce it from the fact that 6-valent vertices only join adjacent colors, and 4-valent vertices join only distant colors.
In addition to the bimodules B i above, we will require the use of the bimodule R ⊗ R i,i+1 R{−3}, where R i,i+1 is the ring of invariants under the transpositions (i, i+1) and (i+1, i+2), and will use a black squiggly line, as in equation 7 below, to represent it. This bimodule comes into play in the isomorphisms and which we will use in the proof of Reidemeister move III. As usual in a diagrammatic category, composition of morphisms is given by vertical concatenation, and the monoidal structure is given by horizontal concatenation. We then allow Z-linear sums of graphs, and apply relations to obtain our category SC 1 . The reations come in three flavors: one color, two distant colors, two adjacent and one distant, and three mutually distant colors. We do not list all of them here, just the consequences necessary for the calculations at hand, and refer the reader to [4] and [11] for a complete picture. Our graphs are invariant under isotopy and in addition we have the following isomorphisms or "decompositions": (6) Note that this relation is precisely that of 1 described diagrammatically.

(7)
Here we have the graphical counterpart of 4 and 5.
Remark. There is a functor from this graphical category to the category of R-bimodules, sending a line colored i to B i and each generator to an appropriate bimodule map. The functor gives an equivalence of categories between this diagrammatic category and the subcategory SC 1 of R-bimodules mentioned in the previous section, so the use of the same name is legitimate.
Our diagrammatic category has many wonderful properties, such as the self-adjointness of B i , which permits us to "twist" morphisms around and view any morphism as one from or to the empty diagram. This allows for a very hands-on, explicit, understanding of hom-paces between objects in SC 1 , which was key in proving functoriality in [5].
Primarily we will work in another category denoted SC 2 , the category formally containing all direct sums and grading shifts of objects in SC 1 , but whose morphisms are forced to be degree 0. In addition, we let SC be the Karoubi envelope, or idempotent completion, of the category SC 2 . Recall that the Karoubi envelope of a category C has as objects pairs (B, e) where B is an object in C and e an idempotent endomorphism of B. This object acts as though it were the "image" of this projection e, and in an additive category behaves like a direct summand. For more information on Karoubi envelopes, see Wikipedia. It is really here that the object R ⊗ R i,i+1 R{−3} of 4 and 5 resides. In practice all our calculations will be done in SC 2 , but since this includes fully faithfully into SC they will be valid there as well.

Hochschild (co)homology
Let A be a algebra and M an A-A-bimodule, or equivalently a left A ⊗ A op -module or a right A op ⊗ A-module. The definitions of the Hochschild (co)homology groups HH * (A, M) (HH * (A, M)) are the following: To compute this we take a projective resolution of the A-bimodule A, with the natural left and right action, by projective A-bimodules · · · → P 2 → P 1 → P 0 → 0, and tensor this with M over A ⊗ A op to get The homology of this complex is isomorphic to HH * (A, M).
Example: For any bimodule M, we have where [A, M] is the subspace of M generated by all elements of the form am − ma, a ∈ A and m ∈ M, and M A = {m ∈ M | am = ma for all a ∈ A}. We leave this as an exercise or refer the reader to [6].
If we take the polynomial algebra A = [x 1 , . . . , x n ], with commutative, then we can use a much smaller, "Koszul," resolution of A by free A ⊗ A-modules. This is gotten by taking the tensor product of the following complexes This resolution has length n, and its total space is naturally isomorphic to the exterior algebra on n generators tensored with A ⊗ A. Hence, we get that the Hochschild homology of a bimodule M over A is made up of 2 n copies of M, with the differentials coming from multiplication by where Z[I] is the rank 1 free abelian group generated by the symbol [I] (i.e. it's there to keep track where exactly we are in the complex). Here, the differential takes the form and the sign is taken as negative if I contains an odd number of elements less than i.
Remark. For the polynomial algebra, the Hochschild homology and cohomology are isomorphic, for any bimodule M. This comes from self-duality of the Koszul resolution for such algebras. Hence, we will be free to use either homology or cohomology groups in the constructions below.
For us, taking Hochschild homology will come into play when looking at closed braid diagrams. To a given resolution of a braid diagram we will assign a Soergel bimodule; "closing off" this diagram will correspond to taking Hochschild homology of the associated bimodule. More details of this below in section 3.2.

.1 The matrix factorization construction
As stated above we will work with Z-graded, rather than Z/2Z-graded, matrix factorizations and follow closely the conventions laid out in [17]. We begin by first assigning the appropriate complex to a single crossing and then extend this to general braids.
Gradings: Our complex will be triply graded, coming from the internal or "quantum" grading of the underlying ring, the homological grading of the matrix factorizations, and finally an overall homological grading of the entire complex. It will be convenient to visualize our complexes in the plane with the latter two homological gradings lying in the horizontal and vertical directions, respectively. We will denoted these gradings by (i, j, k) = (q, 2gr h , 2gr v ) and their shifts by curly brackets, i.e. {a, b, c} will indicate a shift in the quantum grading by a, in the horizontal grading by b, and in the vertical grading by c. Note that following the conventions in [17] we have doubled the latter two gradings. Consider the two types of crossings D + and D − , as in figure 1, with outgoing edges labeled by k, l, and incoming edges labelled by i, j . Let be the underlying ring associated to a crossing. To the positive crossing D + we assign the following complex: To the negative crossing D − we assign the following complex: O O A few useful things to note: The horizontal and vertical differentials d + and d v are homogeneous of degrees (2, 2, 0) and (0, 0, 2), respectively. For those more familiar with [11] and hoping to reconcile the differences, note that in R c multiplication by , so up to some grading shifts we are really working with the same underlying complex as in the original construction, but of course now over Z, not Q.
To write down the complex for a general braid we tensor the above for every crossing, keeping track of markings, and then replace the underlying ring with a copy of the edge ring R(D). More precisely, given a diagram D of a braid let where w and b are the writhe and the number of strands of D, respectively.
Remark. In [17], this is what J. Rasmussen calls the "middle HOMFLY homology." The relation between this link homology theory and the HOMFLY-PT polynomial is that for any The reduced complex: There is a natural subcomplex C(D) ⊂ C(D) defined as follows: let R(D) ⊂ R(D) to be the subring generated by x i − x j where i, j run over all edges of D and let C(D) be the subcomplex gotten by replacing in C(D) each copy of R(D) by one of R(D). A quick glance at the complexes C(D + ) and C(D − ) will reassure the reader that this is indeed a subcomplex, as the coefficients of both d v and d + lie in R(D). We will refer to C(D) as the reduced complex for D.
• If i is an edge of D we can also define the complex C(D, i) := C(D)/(x i ). It is not hard to see that C(D, i) ∼ = C(D) and is, hence, independent of the choice of edge i. See [17] section 2.8 for a discussion as well as [11].
Below we will work primarily with the reduced complex C(D), and will stick with the grading conventions of [17], which are different than that of [11].
where w and b are the writhe and the number of strands of D, respectively.

Remark. For any link
We can look at the complex C(D) in two essential ways: either as the tensor product, over appropriate rings, of C(D + ) and C(D − ) for every crossing in our diagram D (as described above), or as a tensor product of corresponding complexes over all resolutions of the diagram. Although this is really just a matter of point of view, the latter approach is what we find in the original construction of Khovanov and Rozansky, as well as in the Soergel bimodule construction to be described below. To clarify this approach, consider the oriented D o and singular D s resolution of a crossing as in diagram 1. Assign to D o the complex where the maps are given by d v as defined above. [For simplicity we leave out the internal grading shifts.] Let a resolution of a link diagram D be a resolution of each crossing in either of the two ways above, and let the complex assigned to each resolution be the tensor product of the corresponding complexes for each resolved crossing. Then, modulo grading shifts, our total complex can be viewed as where D res is the diagram of a given resolution. This closely mimics the "state-sum model" for the Jones polynomial, due to Kauffman [7], or the MOY calculus of [14] for other quantum polynomials.

The Soergel bimodule construction
We now turn to the Soergel bimodule construction for the HOMLFY-PT homology of [9]. Recall from section 2.2 that the Soergel bimodule . . , x n−1 − x n ] is the ring generated by consecutive differences in variables x 1 , . . . , x n (n is the number of strands in the braid diagram), and R i ⊂ R is the subring of S 2 -invariants corresponding to the permutation action We place B i in homological grading 0 and increase/decrease by 1, i.e. in the complex for D + , R{2} is in homological grading −1. Note, this grading convention differs from [9], and is the convention used in [5]. The complexes above are known as Rouquier complexes, due to R. Rouquier who studied braid group actions with relation to the category of Soergel bimodules; for more information we refer the reader to [5], [9], and [18].

Figure 3:
Given a braid diagram D we tensor the above complexes for each crossing, arriving at a total complex of length k, where k is the number of crossings of D, or equivalently the length of the corresponding braid word. Each entry in the complex can be thought of as a resolution of the diagram consisting of the tensor product of the appropriate Soergel bimodules. For example, to the graph in 3.2 we assign the bimodule B 1 ⊗ B 2 ⊗ B 1 . That is, modulo grading shifts, we can view our total complex as C(D) = resolutions C(D res ).
To proceed, we take Hochschild homology HH(C(D res )) for each resolution of D and arrive at the complex HH(C(D)) = resolutions HH(C(D res )), with the induced differentials. Finally, taking homology of HH(C(D)) with respect to these differentials gives us our link homology.
Definition 6. {reduced homology} Given a braid diagram D of a link L we define its reduced HOMFLY-PT homology to be the group H(HH(C(D))).
Of course, now that we have defined reduced HOMFLY-PT homology in two different ways, it would be nice to reconcile the fact that they are indeed the same. Proof. The proof in [9] works without any changes for matrix factorizations and Soergel bimodules over Z. We sketch it here for completeness and the fact that we will be referring to some of its details a bit later. Lets first look at the matrix factorization C(D s ) (unreduced version) associated to a singular resolution D s . Now C(D s ) can be though of as a Koszul complex of the sequence ( . Now this sequence is regular and the complex has cohomology in the right-most degree. The cohomology is the quotient ring This is naturally isomorphic to the Soergel bimodule B ′ i (notice that this is the "unreduced" Soergel bimodule) over the polynomial ring Z[x i , x j ]. The left and right action of R ′ on B ′ i corresponds to multiplication by x i , x j and x k , x l , respectively. Quotienning out by Now lets consider a general resolution D res . The matrix factorization for D res is, once again, just a Koszul complex corresponding to a sequence of two types of elements. The first ones are as above, i.e. they are of the form x k + x l − x i − x j and x k x l − x i x j and come from the singular resolutions D s , and the remaining are of the form x i − x j that come from "closing off" our braid diagram D, which in turn means equating the corresponding marks at the top and bottom the diagram. Now it is pretty easy to see that the polynomials of the first type, coming from the D s 's form a regular sequence and we can quotient out by them immediately, just like above. The quotient ring we get is naturally isomorphic to the Soergel bimodule B ′ (D res ) associated to the resolution D res . At this point all we have left is to deal with the remaining elements of the form x i − x j coming from closing off D; to be more concrete, the Koszul complex we started with for D res is quasi-isomorphic to the Koszul complex of the ring B ′ (D res ) corresponding to these remaining elements. This in turn precisely computes the Hochschild homology of B ′ (D res ).
Finally if we downsize from B ′ i to B i and from C(D res ) to C(D res ) we get the required isomorphism. For more details we refer the reader to [9]. ⋄ Gradings et all: We come to the usual rigmarole of grading conventions, which seems to be evepresent in link homology. Perhaps when using the Rouquier complexes above we could have picked conventions that more closely matched those of 3.1. However, we chose not to for a couple of reasons: first there would inevitably be some grading conversion to be done either way due to the inherent difference in the nature of the constructions, and second we use Rouquier complexes to aid us in just a few results (namely the proof of Reidemeister moves II and III), and leave them shortly after attaining these; hence, it is convenient for us, as well as for the reader familiar with the Soergel bimodule construction of [9] and the diagrammatic construction of [4], to adhere to the conventions of the former and the subsequent results in [5]. For completeness, we descibe the conversion rules. Recall that in the matrix factorization construction of 3.1 we denoted the gradings as (i, j, k) = (q, 2gr h , 2gr v ).
• To get the cohomological grading in the Soergel construction take (j − k)/2 from 3.1.
• The Hochschild here matches the "horizontal" or j grading of 3.1.
• To get the "quantum" grading i of 3.1 of an element x, take Hochschild grading of x minus deg(x), i.e. deg(x) = j(x) − i(x).

Diagrammatic Rouquier complexes
We now restate the last section in the diagrammatic landuage of [5] as outlined above in 2.2.
The main advantage of doing this is the inherent ability of the graphical calculus developed by Elias and Khovanov in [4] to hide and, hence simplify, the complexity of the calculations at hand. Recall that we work in the integral version of Soergel category SC 2 as defined in section 2.3 of [5], which allows for constructions over Z without adjoining inverses (see section 5.2 in [5] for a discussion of these facts). Recall, that an object of SC 2 is given by a sequence of indices i , visualized as d points on the real line and morhisms are given by pictures or graphs embedded in the strip R × [0, 1]. We think of the indices as "colors," and depict them accordingly. The Soergel bimodule B i is represented by a vertical line of "color" i (i.e. by the identity morphism from B i to itself) and the maps we find in the Rouquier complexes above, section 3.2, are given by those referred to as "start-dot" and "end-dot." More precisely, the complexes C(D − ) and C(D + ) become For completeness and ease we remind the reader of the diagrammatic calculus rubric used to contruct Rouquier complexes for a given braid diagram.

Conventions
We use a colored circle to indicate the empty graph, but maintain the color for reasons of sanity. It is immediately clear that in the complex associated to a tensor product of d Rouquier complexes, each summand will be a sequence of k lines where 0 ≤ k ≤ d (interspersed with colored circles, but these represent the empty graph so could be ignored). Each differential from one summand to another will be a "dot" map, with an appropriate sign.
1. The dot would be a map of degree 1 if B i had not been shifted accordingly. In SC 2 , all maps must be homogeneous, so we could have deduced the degree shift in B i from the degree of the differential. Because of this, it is not useful to keep track of various degree shifts of objects in a complex. Hence at times we will draw all the objects without degree shifts, and all differentials will therefore be maps of graded degree 1 (as well as homological degree 1). It follows from this that homotopies will have degree -1, in order to be degree 0 when the shifts are put back in. One could put in the degree shifts later, noting that B ∅ always occurs as a summand in a tensor product exactly once, with degree shift 0.
2. We will use blue for the index associated to the leftmost crossing in the braid, then red and dotted orange for other crossings, from left to right. The adjacency of these various colors is determined from the braid.
3. We read tensor products in a braid diagram from bottom to top. That is, in the following diagram, we take the complex for the blue crossing, and tensor by the complex for the red crossing. Then we translate this into pictures by saying that tensors go from left to right. In other words, in the complex associated to this braid, blue always appears to the left of red.
4. One can deduce the sign of a differential between two summands using the Liebnitz rule, d(ab) = d(a)b + (−1) |a| ad(b). In particular, since a line always occurs in the basic complex in homological dimension ±1, the sign on a particular differential is exactly given by the parity of lines appearing to the left of the map. For example, 5. When putting an order on the summands in the tensored complex, we use the following standardized order. Draw the picture for the object of smallest homological degree, which we draw with lines and circles. In the next homological degree, the first summand has the first color switched (from line to circle, or circle to line), the second has the second color switched, and so forth. In the next homological degree, two colors will be switched, and we use the lexicographic order: 1st and 2nd, then 1st and 3rd, then 1st and 4th... then 2nd and 3rd, etc. This pattern continues.

Checking the Reidemeister moves
We will use the matrix factorization construction of section 3.1 to check Reidemeister move I, as it is not very difficult to verify even over Z that this goes through, and the diagrammatic calculus of section 3.2.1 for the remaining moves. There are two main reasons for the interplay: first, checking Reidemeister II and III over Z using the matix factorization approach is rather computationally intensive (it was already quite so over Q in [11] with all the algebraic advantages of working over a field at hand); second, at this moment there does not exist a full diagrammatic description of Hochschild homology of Soergel bimodules, which prevents us from using a pictorial calculus to compute link homology from closed braid diagrams. Of course, for Reidemeister II and III we could have used the computations of [5], where we prove the stronger result that Rouquier complexes are functorial over braid cobordisms, but the proofs we exhibit below use essentially the same strategy as the original paper [11], but are so much simpler and more concise that they underline well the usefulness of the diagrammatic calculus for computations. With that said, we digress... A small lemma from linear algebra, which Bar-Natan refers to as "Gaussian Elimination for Complexes" in [1], will be very helpfup to us. Lemma 8. If φ : B → D is an isomorphism (in some additive category C), then the four term complex segment below is isomorphic to the (direct sum) complex segment Both of these complexes are homotopy equivalent to the (simpler) complex segment Here the capital letters are arbitrary columns of objects in C and all Greek letters are arbitrary matrices representing morphisms with the appropriate dimensions, domains and ranges (all the matrices are block matrices); φ : B → D is an isomorphism, i.e. it is invertible.
Proof: The matrices in complexes (1) and (2) differ by a change of bases, and hence the complexes are isomorphic. (2) and (3) differ by the removal of a contractible summand; hence, they are homotopy equivalent.
Up to homotopy, the right-hand side of the complex dissapears and only the top left corner survives after quotioning out by the relation x 2 − x 1 . Note that the overall degree shifts of the total complex make sure that the left-over entry sits in the correct tri-grading.
Similarly, the complex C(D I b ) for the left-hand side braid in Reidemester Ib, has the form The left-hand side is annihilated and the upper-right corner remains modulo the relation

Reidemeister II
Proof. Lets first consider the braid diagrams for Reidemeister type IIa in figure 5 in SC 2 and its pictorial counterpart 6. The complex we are interested in is Inserting the decomposed B i ⊗B i and the corresponding maps, we find two isomorphisms staring at us; we pick the left most one and mark it for removal.  With the marked acyclic subcomplex removed, we arrive at our desired result, the complex assigned to the no crossing braid of two strands as in figure 5. The computation for Reidemeister IIb is virtually identical. ⋄

Reidemeister III
Proof. Luckily, we only have to check one version of Reidemeister move III, but as the reader will see below even that is pretty easy and not much harder than that of Reidemeister II above. We follow closely the structure of the proof in [11], utilizing the bimodule R ⊗ R i,i+1 R{−3} and decomposition 7 to reduce the complex for one of the RIII braids to that which is invariant under the move or, equivalently in our case, invariant under color flip. We start with the braid on the left-hand side of III in figure 5; the corresponding complex, with decomposition 6 and 7 given by dashed/yellow arrows, is We insert the decomposed bimodules and the appropriate maps; then we change bases as in Lemma 8 (the higher matrix of the two is before base-change, and the lower is after). We strike out the acyclic subcomplex and mark another one for removal; yet again we change bases (the lower matrix is the one after base change). Figure 11: Reidemeister III complex, with another acyclic subcomplex marked for removal Now we are almost done; if we can prove that the maps are invariant under color change, we would arrive at a complex that is invariant under Reidemeister move III. To do this we must stop for a second, go back to the source and examine the original, algebraic, definitions of the morphisms in [4]; upon doing so we are relieved to see that the maps we are interested in are actually equal to zero (they are defined by sending 1 . In all, we have arrived at Figure 12: Reidemeister III complex -the end result, after removal of all acyclic subcomplexes Repeating the calculation for the braid on the right-hand side of RIII, figure 5, amounts to the above calculation with the colors switched -a quick glance will convince the reader that the end result is the same complex rotated about the x-axis. ⋄

Observations
Having seen this interplay between the different constructions, perhaps it is a good moment to highlight exactly what categories we do need to work in so as to arrive at a genuine link invariant, or a braid invariant at that. Well, let us start with the latter: we can take the category of complexes of Soergel bimodules KOM(SC) (either the diagrammatic or "original" version) and construct Rouquier complexes; if we mod out by homotopies and work in KOM h (SC), we arrive at something that is not only an invariant of braids but of braid cobordisms as well (over Z or Q if we wish). Now if we repeat the construction in the category of complexes of graded matrix factorizations KOM(mf ), we have some choices of homotopies to quotient out by. First, we can quotient out by the homotopies in the category of graded matrix factorizations and work in KOM(hmf ) and second, we can quotient in the category of the complexes and work in KOM h (mf ), or we can do both and work in KOM h (hmf ). It is immediate that working in KOM h (mf ) is necessary, but one could hope that it is also sufficient. A close look at the argument of Claim 7, where the two constructions are proven equivalent, shows that if we start with the Koszul complex associated to the resolution of a braid D res the polynomial relations coming from the singular vertices in D res form a regular sequence and, hence, the homology of this complex is the quotient of the edge ring R(D res ) by these relations and is supported in the right-most degree. It is this quotient that is isomorphic to the corresponding Soergel bimodule, i.e. the Koszul complex is quasiisomorphic, as a bimodule, to B ′ (D res ). Hence, we really do need to work in KOM h (hmf ), to have a braid invariant or an invariant of braid cobordisms, or a link invariant.
Anyone, who has suffered throught the proofs of, say, Reidemeister III in [11] would probably find the above a relief. Of course, much of the ease in computation using this pictorial language is founded upon the intimate understanding and knowledge of hom spaces between objects in SC, which is something that is only available to us due to the labors Elias and Khovanov in [4]. With that said, it would not be suprising if this diagrammatic calculus would aid other calclulations of link homology in the future.
All in all we have arrived at an integral version of HOMFLY-PT link homology; combining with the results of [5] we have the following: Theorem 9. Given a link L ⊂ S 3 , the groups H(L) and H(L) are invariants of L and when tensored with Q are isomorphic to the unreduced and reduced versions, respectively, of the Khovanov-Rozansky HOMFLY-PT link homology. Moreover, these integral homology theories give rise to functors from the category of braid cobordisms to the category of complexes of graded R-bimodules. 5 Rasmussen's spectral sequence and integral sl(n)-link homology It is time for us to look more closely at Rasmussen's spectral sequence from HOMFLY-PT to sl(n)-link homology. For this we need an extra "horizontal" differential d − in our complex, and here is the first time we encounter matrix factorizations with a non-zero potential; as before, to a link diagram D we will associate the tensor product of complexes of matrix factorizations with potential for each crossing. These will be complexes over the ring with total potential where the p(x) ∈ Z[x]. We do not specify the potential p(x) at the moment as the spectral sequence works for any choice; later on when looking at sl(n)-link homology we will set p(x) = x n+1 . To define d − , let p i = W p /(x k − x i ) and p ij = −W p /(x k − x i )(x k − x j ) (recall that in R c , (x k − x i )(x k − x j ) = x i x j − x k x l , and note that these polynomials are actually in R c ).
To the positive crossing D + we assign the following complex: To the negative crossing D − we assign the following complex: The total complex for a link L with diagram D will be defined analagously to the one above, i.e. The main result of [17] is the following: