Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants

We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky (KR) polynomials, simultaneously for all values of N. No matrix factorization and related tedious calculations are needed in such approach, which can therefore become not only conceptually, but also practically useful.


Introduction
The theory of knot polynomials [1,2] is nowadays one of the fast-developing branches of theoretical and mathematical physics, unifying critical ideas from many different other subjects, from topology to localization and AGT relations. A special part of the story is relation to index theorems and homological algebra. In physical language this is the possibility to reinterpret the averages of characters in Chern-Simons theory (HOMFLY polynomials) as Euler characteristics of certain complexes, invariant under infinitesimal deformations of curves, and further promote them to Poincare polynomials of the same complexes in a way, which preserves the invariance. Poincare polynomial is a generating function of cohomologies, therefore the task is to use topological invariance to reduce the functional integral to the infrared -to the zero-modes of differential operators, which can be rewritten as acting on parameters ("times") of the low-energy effective action. Remarkably, these operators have a typical form of cut-and-join operators -or, what is nearly the same, of the Hamiltonians of integrable systems.
We reviewed the main technical parts of this Khovanov's categorification program [3]- [22] in [23,24] with the main example of GL(2) gauge group and the fundamental representation. This theory of Jones superpolynomials is very transparent and clear, especially after the pedagogical presentation of D.Bar-Natan [4] and following advances in computerization. The problem is, however, far more severe for higher GL(N ) groups, where the main results are obtained with the help of Khovanov-Rozansky construction [7], based on the additional technique of matrix factorization, which makes the story obscure both conceptually and technically. We return to this construction -quite beautiful by itself -in one of the papers of our review series, but before that we prefer to present a natural generalization of the N = 2 story, immediately implied by the tensor algebra approach, which was our starting point in [23], and without any direct reference or use of matrix factorization.
Technically the peculiarity of N = 2 was that representations of SL(2) are real, and this allowed to substitute the naturally appearing cycle decomposition, true for all N , by that into planar cycles -and such construction seemed and was un-generalizable to arbitrary N . However, as we show in the present paper, the naive one [23], with non-planar cycles, actually reproduces all the results for N = 2 -and works just the same way for all values of N . Moreover, it directly provides the answers as explicit functions of N . We restrict in this paper to the simplest examples, and reproduce just the very first items of the currently available list of Khovanov-Rozansky polynomials (both reduced and unreduced) in the fundamental representation, worked out by terribly complicated calculations in [19]. 1 Our feeling is that with the alternative technique, suggested in the present paper, the list can be reproduced and substantially enlarged -to the same extent as it is available for Jones superpolynomials in [30].

From knots to knot diagrams and tensor algebras
The very first step in the theory of knot polynomials is to reformulate the problem in terms of knot diagrams -the graphs with colored vertices. In knot theory the graphs (i) are planar, (ii) have vertices of valence (2, 2), (iii) just two colors are allowed. The tensor-algebra construction in the style of [31], which we are going to use is in no way restricted by these choices -still in the present paper we discuss this standard setting, and some technicalities will depend on it.
For the way to reduce to this form the more conventional formulations of the problem -either in terms of knot theory or in those of Chern-Simons correlators in temporal gauge -see [32] and references therein. In one word, knot diagrams (planar graphs) appear when oriented lines in 3 dimensions (oriented knots or links) are projected on 2-dimensional plane. Not to loose information, one should distinguish which of the two lines was above another, when their projections cross -this means that there are two different types of vertices, which we call black and white (in R-matrix formalism they would be associated with R and R −1 ). To keep topological invariance -equivalence of different projections of the same link/knot -one should consider only Redemeister-invariant functions on the graphs. 1 While knowing ordinary knot polynomials for generic N immediately provides HOMFLY polynomials, depending on A = q N instead of N , the story is more complicated in the case of superpolynomials. The basic difference is that Khovanov-Rozansky polynomials depend on the quantum numbers [N − k] = Aq −k −A −1 q k q−q −1 and therefore themselves are not Laurent polynomials of A with positive coefficients, as superpolynomials are requested to be (at least in the fundamental representation). In fact, the N -dependent Khovanov-Rozhansky polynomials live in a seemingly non-trivial factor-space of the A-dependent superpolynomialsand the lifting to superpolynomials can be a little tricky [25]- [29]. However, the study of this lifting is still severely restricted by the lack of diverse examples -this adds to the need of developing technical means to effectively produce arbitrary Khovanov-Rozansky polynomials.
Our starting point is just the theory of (2, 2)-valent planar graphs D c with vertices of two kinds (colors), and our main claim is that the "physical input" implies simply the need to consider invariant tensors of the tensor algebra T N -this condition alone will lead us to Redemeister-invariant knot polynomials. Exactly like in the case of N = 2, for all N the HOMFLY invariants will just count the number of cycles in the resolution of knot diagram (which however, still need to be properly defined), while Khovanov-Rozansky ones will count cohomologies (Poincare polynomials) of associated complexes, made in a nearly canonical way from the vector spaces (actually, for generic N these are factor-spaces). HOMFLY are their Euler characteristics, and depend only on dimensions of the vector spaces, not on the morphisms between them, and these dimensions are just made from the (graded) numbers of above-mentioned cycles. In other words, direct application of tensor-algebra ideas a la [31] provides a natural but previously unnoticed construction of the commutative quiver on the hypercube, involving vector spaces V = C N of arbitrary dimension N , so that the properly normalized Poincare polynomial of associated quiver complex reproduces Khovanov-Rozansky polynomials. This brief description implies the familiarity with either [4] or [23,24] -for the sake of completeness we repeat that standard construction in the case of N = 2 in section 3 below, while now we return to tensor algebra.
If no other structures are introduced, the tensor algebra T N itself has just two SL(N )-invariant tensors: The covariant i1. ..i N is made out of those two. The fact that there are exactly two invariant tensors appears to match perfectly with the desire to have vertices of exactly two kinds (colors). But to make the contact, we need first to get the proper valences. Valence (2, 2) means that we need tensors with two upper and two lower indices. Clearly, there are exactly three options: The first two of them are planar, the third is not -this is the reason why for N = 2 one uses the linear combinations of δ i k δ j l and ij kl . Still this choice is not so obvious. For integer N (or in the case of no qdeformation, if one prefers this language) the three structures are linearly dependent: For q = 1 things are not so simple, (see sec. 5.5.3 below), still the dilemma of which two of the three vertices to choose remains.
The conventional approach is to take δ i k δ j l and ij kl for N = 2, this decomposes resolved knot diagrams into units of planar cycles -but is well known not to work for N = 2 (does not give anything besides Jones polynomials, at best). Starting from sec. 5, we switch to alternative choice: Now the cycles are not all planar, moreover some of them enter with negative signs, but instead the construction appears to work not only for N = 2, but for arbitrary N . "Works" means that it provides Redemeister-invariant answers, which depend non-trivially on N , moreover, these answers coincide with those from [19], derived with the help of the standard matrix-factorization-induced Khovanov-Rozansky construction.
The plan of this paper follows as close as possible the main logic of Khovanov's approach: HOMFLY polynomial q-graded vector H (Dc) (N |q), factor-spaces, sec.6 sec.5 (HOMFLY) counting cycles associated with cyclvaes ↑ ↓ sec. 6 q-Euler characteristic linear maps (morphisms) sec.6 of K(D c ) between vector spaces, making from H(D c ) a commutative quiver ↓ associated complex K(D c ) sec.6 ↓ Khovanov-Rozansky polynomial P (Dc) (N |q|T ), sec.6 counting its cohomologies Section 7 describes the first steps towards a similar systematization of the results of sec. 6 for KR superpolynomials, but this story has more subtleties and interesting deviations -it will be continued in more detail elsewhere.
The "global" approach, suggested in the present paper is to follow the chain: As presented in this paper, our construction is not fully algorithmic. The two points, where some art is applied, are the quantization of dimensions -here one can control the choice by comparison with the HOMFLY polynomials -and adjustment of morphisms: at this stage we use a very appealing "maxmal-subtraction" rule. We also do not fully prove here the Redemeister invariance. It does not look too difficult to formalize all these details, but our goal in this paper is rather to demonstrate the spirit of our radical modification of Khovanov-Rozansky approach and its impressive effectiveness and simplicity in concrete examples.
3 Basic ideas in the case of N = 2 and beyond We begin by describing the general ideas of Khovanov approach, following [4] and [23,24].

Hypercube H(D) of colorings
Consider not just a given link diagram D c , but the whole set with all possible colorings, i.e. the given graph D with all possible colors at its vertices. 2 If just two colors are allowed, we number of colorings of the n-vertex graph D is 2 n , and what we get is an n-dimensional hypercube H(D), where each vertex represents one particular coloring c of D: Edges of the hypercube are naturally associated with the elementary flips -inversions of color at one particular vertex of D.
Original link has particular coloring, thus it is associated with one particular "initial" vertex c 0 of the hypercube. Once it is specified, edges become arrows, pointing away from c 0 .
Redemeistermoves are associated with duplication of the hypercube. For example, adding an elementary loop in R1 introduces one extra vertex in D, what implies that the new hypercube consists of two copies of the original one. Similarly, R2 adds two vertices of different color to D -then the new hypercube consists of four copies of the original, while R3 relates the result of adding three vertices to D in two different ways and thus involves a three-dimensional sub-cube in H(D).

Morphisms between the resolutions
The edges of the hypercube connect resolved diagrams of the same topology, but with one vertex resolved differently. In this sense an edge is naturally associated with the morphisms between the two resolutions at one vertex of D.  [23] by the existence of two invariant tensors δ i j and ij in the tensor algebra with N = 2. This choice is difficult to generalize literally to N > 2, though its minor modification is easily generalizable -as we shall see in the next section 5. But first we proceed with the standard approach.
Once resolutions are chosen, the planar graph D c at the vertex c of the hypercube decomposes into ν c disconnected cycles. Thus with each vertex c ∈ H(D) one associates two numbers: this ν c and h c = h 0 (c), which is the distance between c and initial c 0 .
The crucial observation is that the Redemeister moves change ν c and h c in a simple way.
e I q d t The first Redemeister move R1 duplicates the hypercube: H(D) → H(D)∪H (D), so that the corresponding vertices of H (D) have ν c = ν c + 1 and h c = h c + 1. This is because when the white resolution is chosen, the number of cycles does not change at all, while for the black resolution exactly one cycle is added. Now we can easily write down an invariant of R1: Similarly one can check that this is also invariant of R2 and R3.

Towards knot/link polynomials
From here one can go in different directions. One can generalize to other types of graphs (non-planar, nonoriented, with other types of vertices and colorings) -for this one needs to modify the idea to associate colors with resolutions of D and cycles. Instead one can extend the invariant from just a number to function of several variables -i.e. to something closer to the knot polynomials. In what follows we proceed in this second direction. One step seems obvious. Since invariance under R1 above was based on the identity (−1) 1 + (−1) · 2 = 1 (8) it is natural to deform any of the underlined four parameters. Actually, they are all of different nature and can be deformed independently. Since so far we have only one relation (8), one can expect at least three independent deformations. As we shall see, this is the right expectation, but actually it will not be quite so simple to find all the three. The problem is that so far we looked only at R1, moreover, even this we did not do exhaustively. When we wrote that it was true only for inserting an elementary loop with a white vertex, so that its resolution does not add a new disconnected cycle. However, the resolution of the black vertex does the opposite: adds the cycle and increases ν c by one. when we insert the white vertex, the white resolution goes first, and the black one the second -thus we obtain (9). However, if we insert an elementary loop with a black vertex, then black resolution goes first and white resolution second, so that (9) will be substituted by Our invariant now changes sign: moreover this time the relevant identity is slightly different: To get rid of the sign difference we can assume that the overall sign factor in fact counts only white vertices of D, while black ones enter instead with the factor unity, i.e. invariant of both R1 • and R1 • is We also made notation more adequate: invariant depends on the link diagram D and initial coloring c 0 and the "height" h c is counted as the distance from c 0 . It is this quantity that we are going to deform. The most general expression that we can write down, preserving the structure of (13) is The two constraints that we already know state that what defines α • and α • through the other two parameters.

Implication of R2 invariance
The Redemeister move R2 substitutes each vertex of the hypercube H(D) by a square: Only at the vertex • • we have the resolved D with the same set of planar cycles as it had before the Redemeister move. The other three vertices correspond to some other resolution of D -and they should cancel among themselves. This means that now we have two requirements: From (15) and (16) it follows that and this is just a one-parametric family. In order to get more we need to further modify the structure of (14).
5 From 2 to N , HOMFLY polynomials 5.1 Another system of cycles The first step of this modification introduces a new parameter N , such that D = [N ] q -and the question is what should be done with (14) in order to allow such a deformation from N = 2 to arbitrary N . As we already mentioned, our suggestion is to abandon (3) and use (4) instead. This means that instead of the two resolutions at the beginning or s. 4 In fact, this choice seems much more natural from the point of view of the tensor algebra -and it indeed is much easier deformed. The price to pay is that now the second resolution gives rise not only to planar cycles, moreover, it provides not a single cycle, but a linear combination, moreover, with some coefficients negative. When in the next sections 6 we further substitute cycles with vector spaces, this means that some of those will actually be factor-spaces.
As to the present stage, this means that eq. (14) associates not just a single power D νc with each vertex c of the hypercube H(D): when c involves white vertices, there is a linear combination instead. As we shall see below, this actually implies that powers D νc are substituted by less trivial products D (c) of "differentials" which are known to play a big role in other branches of knot theory [25]- [29].
Note that only "negative" differentials appear, reflecting the negative sign in the definition of the white resolution. Also note that, despite there are negative contributions, the total contribution of each vertex c is positive: negative contributions are always smaller than the positive ones. Finally the deformation of (14) which we are going to discuss in this section looks like where D (c) now depends not only on q, but also on additional parameter N (or A). As we shall see, the Redemeister invariance requires that We denote this invariant by H, because it actually is nothing but a HOMFLY polynomial. A priori HOMFLY of a knot is an average of a character (Wilson loop) in Chern-Simons theory [2,32], and -since we consider only knot polynomials in the fundamental representation -at q = 1 it reduces to N . Likewise a link is an averaged product of characters, so that in general (see also eq.(136) or ref. [28], saying that reduced HOMFLY for a knot is always 1 + O(log q), provided A = q N with N fixed, -and unreduced polynomial is N times larger in this limit). This will be always true in our construction.
The rest of this section is just a collection of examples, which tell much more about the story than any formal definitions. Those will be provided elsewhere.

1-dimensional hypercube and the R1-invariance
We begin with the knot diagram D with a single vertex, i.e. of the shape of eight. The corresponding hypercube is one dimensional, i.e. just a segment with two vertices: It is very useful to represent this hypercube as arising in three steps. At the first step we just insert a cross X instead of the true resolution || − X at all white vertices of D c and draw what we call the cycle diagram (boxed in (22) below). The result of || insertion is naturally obtained from X by cutting, and we use the arrow in cycle diagram to show the direction of this cut procedure: in the present case it maps the white vertex into the black one. The vertices where all arrows are only terminating are called drain, and the vertex with all resolutions black, is called the main one, it is always among the drain vertices, and we often put it into a box. Alternatively it could be called Seifert vertex, because the corresponding decomposition is in planar Seifert cycles. At the level of HOMFLY polynomials drain vertices do not play any interesting role, but the Seifert vertex does.
At the second step we construct the "classical" D (c) , which are just the linear combinations of powers D νc : Then we note that they can be naturally rewritten as products, and then, at the third step apply the "obvious" quantization rules for these D (c) : Thus from (19) and (20) we obtain the answers for the single-vertex D with the black and white vertices respectively: i.e. reproduce the HOMFLY polynomial for the unreduced unknot -as they should. Since reduced HOMFLY differ just by division over [N ] we do not consider them separately and from now on denote unreduced HOMFLY by H.
Just the same calculation explains invariance of so constructed HOMFLY polynomials under the first Redemeister move: R1 doubles the hypercube and multiplies the answer by This actually follows from analysis of the following example.

Double eight
Adding one more vertex converts our single-vertex "eight" into the the two-vertex knot diagram D with the shape of a "double eight": Whatever the coloring, this is just an unknot (a result of two applications of R1 ±1 to a circle) -and this is immediately seen from the answers for knot polynomials. Reading from the picture for the hypercube Note that N 3 − N 2 is obtained by subtracting the N ν at the tail of the arrow from N ν at its nose. Likewise, N 3 − 2N 2 + N is the similar alternative-summation along the two paths leading from the given vertex to main one (boxed), where all vertices are black and all resolutions are trivial (it corresponds to decomposition of D intp Seifert cycles). Now we apply the obvious quantization rules and get the HOMFLY polynomials (A = q N ):

Hopf link and the R2-invariance:
The simplest next example is the Hopf link. The knot diagram has two vertices, the hypercube has 2 2 = 4 vertices, like in the case of the double eight, i.e. it is again the 2-dimensional square (or rhombus): In this example there are two drain vertices in the hypercube. Hopf link per se corresponds to choosing as initial one hypercube vertex with two identical colors, e.g. the main vertex •• at the bottom. Then In terms of A = q N this is This is the right answer for the HOMFLY polynomial. Note that it is reproduced if we accept the quantization Of course, another 2 in (28), which arises just from adding the two identical contributions at vertices •• and •• is not quantized. Taking as initial the white-white vertex we obtain the mirror-symmetric answer: If instead we start from the black-white or white-black vertices the answer will be different -as it should be, because in this case we get the two unlinked unknots: This decomposition into a product of two unknots is the simplest illustration of R2 invariance of our construction. 5.5 Other 2-strand knots and links 5.5.1 Trefoil in the 2-strand realization The first non-trivial knot is the trefoil. It has two standard braid representations: 2-strand and 3-strand. In the 2-strand case the knot diagram D has three vertices, the hypercube H(D) is three-dimensional, with 2 3 = 8 vertices, the cycles diagram is We remind that classical dimensions in the right-hand-side diagram are obtained from the cycles diagram by the simple rule -taking alternated sum along all paths connecting the given vertex with the boxed main one: The knot polynomial, obtained by our rules is: what is the standard answer. Note that it is obtained, if in the www vertex we use the following quantization For another coloring bbw we have instead: The same unknot will be obtained, if initial vertex is bww.
For www the answer is mirror-symmetric trefoil: Coincidence with the unknot is guaranteed by the right quantization rule 4 −→ [2] 2 , thus one can say, that this rule is derived from the Redemeister invariance.

Generic knot/link [2, k]
Unknot Hopf link and the trefoil are the members of entire series of k-folds -the closures of a 2-strand braid.
It is instructive to perform our calculation for entire series at once. The cycle diagram is actually a sequence i.e. consists of alternated two-and -single cycle vertices taken with the multiplicities C j k and connected by arrows, which form the k-dimensional hypercube. All vertices with 2 cycles are drain. The corresponding classical hypercube is and the quantization prescription, validated by the known answer for the HOMFLY polynomial and/or the Redemeister invariance is . . .
This is indeed the same as the well known [34]- [40] 1

Towards Kauffman-like formalism
The R-matrix approach [32], [41]- [46] is to simply write down explicit matrices at place of vertices of the link diagram D c0 : Redemeister invariance is guaranteed by the properties R1 : plus various permutations and inversions. As a generalization of (3) and (4), where X is a graded version of contraction of two -tensors.
The N = 2 version of this construction (Kauffman's R-matrix [33]) is presented in detail in sec.1 of [23], where afterwards numerous examples are considered (see also sec.4 of [28] for more advanced applications). Specifics of N = 2 was that one could actually deal with ordinary δ and tensors, and q can be introduced only in traces, by "analytic continuation" from D = 2 to D = [2] = q + q −1 . For general N such simple approach does not seem to work.
Still, if one allows to q-deform -tensors, the situation is not so pessimistic. Here we just report a few simple observations, relevant for the case of the 2-strand knots, which can imply that some kind of generalization to arbitrary N can still be possible.
When N = 2, we can consider a rank (2, 0) tensor˜ with components and as its dual tensor˜ * of rank (0, 2) with:˜ Then, the "vertex" tensor of rank (2, 2) Passing to N = 3 we can take a rank (3, 0) tensor˜ with non-vanishing components and its dual of rank (0, 3) tensor˜ * with: Then for the 6 ways of getting a scalar from the pair˜ ,˜ * we have: Clearly, the contraction corresponding to the "eight" graph is: ijk ijk = [3] [2]. For the (2, 2) tensor X we also have a number of choices: For a pair of rank (2, 2) tensors X, Y denote by X * Y = X rs ij Y kl rs their straightforward multiplication of rank (2, 2). Then Generalization to higher N is straightforward. Clearly, contractions of the q-deformed tensors are capable to reproduce the peculiar structures [2] n [N ][N − 1], providing our quantities D (c) , at least for the 2-strand knots. A question is, however, if one can make these observations into a working formalism, which would not just coincide with the standard quantum-R-matrix technique [41,43], using explicitly the additional Lie-algebrainduced structure on the tensor algebra. 5.6 Trefoil and the figure-eight knot in the 3-strand realization The 3-strand braid with four vertices, depending on the coloring, describes both the trefoil 3 1 (if all the four vertices are black or all white) and the figure-eight knot 4 1 (if colors are alternating).
We do not show the arrows, which form the 4d hypercube. Clearly there are three drain vertices, each with 3 cycles.
The classical dimensions are given by the general rules (one should only imagine the right configuration of arrows, suppressed in our diagrams): It follows, that what is indeed the right answer for the trefoil, coinciding with (33). All quantizations are obvious, except for the factor in the box. If we quantize then the 3-strand and 2-strand expressions for the trefoil 3 1 are related by thus above quantization rule can be justified by the Redemeister invariance. Also obvious are the degeneracies bbbw = bbwb = bwbb = wbbb, bwbw = wbwb etc.
For alternating colors -and the same expression for quantum dimensions at the hypercube vertices -we get the right expression for HOMFLY of the figure-eight knot 4 1 : Finally, two other types of colorings provide unknots:

Twist knots
Twist knots is in a sense the simplest 1-parametric family (see, for example, sec.5.2 of [28]), which includes unknot, trefoil and the figure-eight knot 4 1 . They are made out of the 2-strand braid, only -in variance with the torus knots -anti-parallel: Here k can be both positive and negative. If the number of crossing in the antiparallel braid is odd, this changes orientation at the two-vertex "locking block". The corresponding knot diagrams (after rotation by 90 • ) are:  Still, the number of cycles in both cases is p + 1, so that in both cases the hypercube vertex b p+2 contributes When some vertex is changed from black to white, one subtracts a contribution with a crossing at this vertex, what changes the number of cycles: for example, when there is just one white vertex, subtraction contains p cycles, and the contribution of b p+1 w vertex in the hypercube is When all vertices are of the same color then the knot is (p + 1) 2 for even p and (p + 2) 2 for odd p. If the two vertices at the top (two "horizontal" vertices) have the opposite color to the p vertical ones, then the knot is (p + 2) 1 for even p and (p + 1) 1 for odd p. When the two horizontal vertices are of different colors, we get an unknot. If some vertical vertices have different colors, what matters is their algebraic sum, The answer for HOMFLY polynomials of the twisted knots is well known, see, for example, sec.5.2 of [28]: For k = 0 and F 0 = 0 we get unknot, for k = 1 and F 1 = −A 2 -the trefoil 3 1 and for k = −1 and F −1 = 1 -the figure eight knot 4 1 . More generally, for positive k we get the knots (2k + 1) 2 , while for negative k -(2 − 2k) 1 in the Rolfsen notation, see [30]. Note that trefoil 3 1 gets its right place in the series of twisted knots, if treated as 3 2 .
Now we proceed to cycle diagrams. In the case of twist knots they have a very special structure. The point is that there are two different types of vertices in D: the two at the "locking block" and p others, located at a vertical axis in above knot diagrams. This implies the obvious block form for the diagrams of cycles and of classical dimensions.
The first table lists  Finally, the last table is that of the quantum (q-graded) dimensions D (c) -in this case they are obtained by the rules, more-or-less familiar from our previous examples. Now we can take different vertices as initial: In the right lower corner we applied the quantization rule (52) -and this provides the right answers: In fact, this is literally the same calculation that we already performed in the previous sec. 5. 6 .
Y 3 in the last line is a deformation (quantization) of (N 2 − 3N + 4), and it turns out to be Indeed. this provides the necessary relations Note that the answer for 5 2 depends on q not only through quantum numbers -and thus is not invariant under the change q −→ q −1 . In fact the complementary Khovanov-Rozansky polynomial H •• = (5 2 |q) exactly by this change.

Generic p
In general we numerate the column in the cycle diagram by i -the number of white vertices among the p vertical ones. Then Note that there is a small difference between odd and even p: it is in the left lower corner of the table. The difference will be more pronounced in the table of dimensions, where it touches all the three entries in the last line.
Actually The table of dimensions can also be immediately written for generic p -moreover, except for the very last line, they are straightforwardly quantized: Slightly non-trivial are only the quantities Y k , defined for odd values k, of which we already know Y 1 = [2] and Since k is odd, these are indeed a polynomials, and they satisfy recursion relation Y cl k+2 − Y cl k = (N − 1) k (N − 2), which is straightforwardly quantized: This quantization rule leads to the standard answers [28] for HOMFLY polynomials of the twisted knots: for odd p and for even p.
Quantization rule (61) −→ (62) can look somewhat artificial. However, as we shall see in the next section, this is not quite true. The gradation-diminishing morphisms are naturally defined for the chains of vector spaces -and this is exactly a structure, implicit in (62).
To finish the entire section 5, devoted to our new HOMFLY calculus, we note that the HOMFLY polynomials are obtained in it by a rather strange two-step procedure: dimensions D (c) are some q-deformed alternated summations over subsets of rhe cycles diagram, and then HOMFLY are alternated sums of these dimension. These two repeated sums can probably be converted into a simpler determinant-like structure, which can also help with the quantization (q-deformation) -like it happens in the studies of closely related [47,48] subject of spin-chain dualities in [49]. This, however, is a subject for a separate investigation.
As to now, we proceed to another deformation -to Khovanov-Rozansky polynomials.
6 Substitute of KR cohomologies 6.1 The idea The main idea of Khovanov's approach is to interpret D νc in (14) as dimensions of q-graded vector spaces V ⊗νc , associated with the vertices of the hypercube H(D), promote the coloring flips at the edges to commuting morphisms between the vector spaces, what converts the hypercube into Abelian quiver. Then with this quiver one associates the complex K(D c0 ), where vector spaces are direct sums of those at vertices of a given height h c − h c0 = i, and differentials d i : C i−1 −→ C i are combinations of commuting morphisms, taken with appropriate signs to ensure the nilpotency d i+1 d i = 0. Then the entire alternated sum (14) can be interpreted as the Euler characteristic of the complex K(D c0 ), while its Poincare polynomial provides a new Redemeister invariant -Khovanov's superpolynomial.
In the language of formulas this means that we first rewrite (14) as where H i = dim q Ker d i+1 Im d i are dimensions of cohomologies (quantum Betti numbers) of the complex K(D c0 ) -and afterwards we promote it to Poincare polynomial depending on additional parameter T , not obligatory equal to −1. Normalization α-parameters can also depend on T , in fact Equivalence between (64) and (65) -the two different representations of the Euler characteristics of a complex -is a simple theorem of linear algebra, which lies in the basement of cohomology theory. It remains true after q-deformation.
Our goal in this section is to explain what happens with this Khovanov's construction when we substitute (14) by the its N -dependent version (19): • First of all, we interpret D (c) as dimensions of some new graded vector spaces, associated with the vertices of the hypercube H(D). In fact, this is the only thing that changes: now the basic vector space V is not two-, but N -dimensional, and D (c) are dimensions of some more sophisticated factor spaces, made from various copies of V . Actually, in the present section we manage without specifying the origin of these spaces explicitly -but for a better grounded approach this should be done, see s.7 below.
• Second, with the edges of H(D) we associate commuting morphisms between these vector spaces. Like in original Khovanov construction, we require that morphisms decrease grading by one. With each edge we associate two morphisms, acting in two directions, both are decreasing. Which morphism actually works, depends on the choice of initial vertex c 0 -all morphism are chosen to point away from c 0 .
• Third, since morphisms are commuting, H(D) c0 has a structure of Abelian quiver -therefore there is an associated complex K(D c0 ). Therefore all the other steps remain the same: Moreover the Poincare polynomial P c 0 (D), introduced in this straightforward way, turns to coincide with the KR superpolynomial, obtained via matrix factorization. . This is the ordinary unknot.

Unreduced superpolynomial
With a single vertex of the hypercube we naturally associate a vector space V = C N with distinguished basis {e 1 , . . . , e N }, V = span(e 1 , . . . , e N ), graded as Thus quantum dimension, which by definition is the Khovanov-Rozansky superpolynomial for the unknot in the fundamental representation is It does not depend on the new parameter T .

Reduced superpolynomial
In the theory of HOMFLY polynomials it is often convenient to divide the answer by HOMFLY for the unknot -what arises is called reduced knot polynomial (and original, undivided, is unreduced). For superpolynomials the procedure is not so innocent: sometime reduced superpolynomial is very different from the unreduced one -and in [19] they are evaluated and listed in separate practically unrelated tables. As reviewed in detail in [24], reduced superpolynomial is obtained in Khovanov approach by the following "reduction" procedure. In the knot diagram D we pick up (mark) one particular edge (in principle, the answer could depend on the choice of this edge, but it does not). Then when at a given vertex v of the hypercube H(D) we decompose D into a set of cycles, we mark the cycles (one per each vertex v ∈ H(D)), and substitute the corresponding vector space V (N -dimensional V = C N in our approach) by a one-dimensional E = C. In the case of unknot this simply means that reduced superpolynomial is unity: Since in this paper we deal only with the fundamental representations we often omit the subscript in what follows. 6.3 Betti numbers from Euler characteristic: naive approach A very naive, still rather powerful approach to evaluation of superpolynomials is to try to saturate the given Euler characteristic by Betti numbers H (c) , which have lower degree in N than original dimensions D (c) .

knot/link
Euler char Euler char Poincare pol = (HOMFLY pol) via D (c) via Betti # s KR superpolynomial . . . Transition between the second and the third columns is just an identity: we rewrite the polynomial in A = q N in the second column as a combination of differentials D −k = {A/q k }/{q} of the minimal possible degreeor, if degree can no longer be diminished, with minimal possible coefficients. Transition from this minimal polynomial to its T -deformed version in the third column is often straightforward -but, strictly speaking, not unique. Fixing this procedure requires explicit definition of morphisms. However, before we pass to them, it is instructive to present the above potentially-ambiguous procedure in one more form. 6. 4 Spaces and morphisms. Unknot as an eight 6.4.1 Listing Within Khovanov approach we should interpret the quantities D (c) from section 5 as dimensions of some vector spaces: . Whatever is the deep origin of these spaces, see s.7 below, knowing D (c) we can list their basis vectors with definite grading degrees. For example: grading unknot for the unknot per se and similarly for the unknot, represented as an eight knot from sec. 5.2: The space V ⊗2 at the vertex • for the 1-fold has dimension [N ] 2 , and there is degeneracy in gradation already within this space: there is just one element of degree 2N − 2, two elements of degree 2N − 4 and so on. Likewise the factor space V ⊗2 /V , which actually stands over the vertex •, has dimension [N ][N − 1], the gradings are odd, there is a single vector of the highest degree is 2N − 3, two of degree 2N − 5 and so on. Listed in the tables are multiplicities of basis vectors of the given gradation degree. At the bottom we write the sums over entire columns, these are quantum (graded) dimensions of the spaces, but below we often use them also to denote the spaces themselves -in cases where this should not make any confusion.

Morphisms and differentials
The table (74) shows very clearly what the decreasing morphisms are: they act along decreasing diagonals -one from left to the right, another from right to the left. The first one has a kernel -its elements of a given grading are obtained by subtracting the multiplicities along the diagonal and the remnants are listed as a column of boxes: clearly the kernel has dimension Similarly the second one has a coimage -again controlled by the algebraic sums along the opposite diagonals: the corresponding deficits are put in double boxes and dimension of coimage is In this particular of the eight knot differentials in the complex K are just the morphisms, therefor from (69) we get: and this demonstrates the Redemeister invariance of the superpolynomial (69).

Reduced case
A similar table and calculation for reduced case are even simpler: where again the boxed and double-boxed entries represent the non-vanishing cohomologies of K(eight c0 ) with c 0 = • and c 0 = • respectively. Therefore reduced superpolynomials are:

Drawing: reduced case
Now we can switch from tables to pictures and draw our two basic decreasing morphisms (in these pictures N = 4, but they can be used to write formulas for arbitrary N ): In other words, π acts as a shift down, accompanied by multiplication by q, so that the grading changes by −1 π : while σ is just multiplication by q −1 : σ : The two complexes, associated with the two initial vertices black (•) and white (•) are: This is a pictorial representation of the The spaces are now "two-dimensional", The complexes this time are: Note that using id ⊗ σ for d • instead of σ ⊗ id would give a wrong answer for CoIm(d • ). 6.5 Example of the 2-foil (Hopf link) 6. 5 In the previous consideration of the eight knot we showed in the same picture the morphisms π and σ acting in different directions -i.e. relevant in the cases of different initial vertices (channels). Now only one channel is represented and the picture shows the complex K(D c0 ) for a given "channel" c 0 = ••. Note that the space C 1 at the second place in the picture consists of two copies of the same rectangular, only one is explicitly shown and 2× is written instead. After that d 2 acts as another shift -this time in the same direction. Because of the conspiracy of gradings the shift itself does not have neither a kernel nor a co-image, when it acts just between the two rectangles. However,it acts on one of the two constituents of C 1 with a plus sign, and withminus -on another: this is the standard way to construct a complex from Abelian quiver. In result, d 2 has a kernel, which is a diagonal subset in the two-constituent C 1 -and this is exactly the image of d 1 . This the cohomology H 1 = 0. As to the target of d 2 , the space C 2 also consists of two constituents, but this time they are not identical, but differ by 2 in grading. d 2 maps C 1 only into the lower constituent, while the upper one remain in co-image -and it forms the cohomology Thus looking at the picture, one straightforwardly concludes that the complex has the Poincare polynomial what reproduces the answer from [19].
This time we showed both constituents of C 1 explicitly. From this picture we immediately read: what differs from the answer q 1−N + q −1−2N T 2 [N − 1] of [19] by a change q −→ 1/q.
The listing of the spaces this time looks as follows: Here we clearly see the advantage of pictures over tables: if we had just the table, we could alternatively box the entire first column, getting alternative expression q N −1 [N ] for the superpolynomial. Knowing the morphisms from the pictures, we can easily reject this option.

Another channel: two unknots. Reduced case
The next exercise is to look at the same link diagram D in another channel, with initial vertex ••. This requires some morphisms, acting in the other direction. It will be a little more convenient to begin from the table. What happens is that now the second column of (87) splits into two, which become the first and the third, while those instead get combined into the second one: This time there is no ambiguity: non-vanishing cohomology lies in the middle column and therefore This is the correct answer: for c 0 = •• we should get a pair of unknots and "reduced" means that one of them is eliminated -thus what we could expect is exactly one unreduced unknot -and this is what we get. The corresponding pattern of morphisms is The differential d 2 annihilates the two components N and

Another channel: two unknots. Unreduced case
In this case we restrict consideration only to the table: In result The picture is also easy to draw, but it gives nothing new and we do not present it here. 6.6 3-foil 6  Cohomology Ker(d 3 )/Im(d 2 ) lies in only one (diagonal) of the three components of Coim(s 2 ) -the rest of it is mapped into the "upper" constituent of the space C 3 , which also contains Coim(d 3 ). Note that d 3 on this component acts in another direction. This allows to make the cohomology smaller, i.e.subtract as much as only possible from coimage of d 3 . This is what we call the maximal subtraction rule.

Morphisms
Morphisms are shown in the picture. One can also write them more formally. For this we introduce the basis {e I } with grad(e I ) = q N +1−2I , I = 1, . . . , N , i = 2, . . . , N . Then Here ω 1 = 1, ω 2 = e 2πi/3 =ω 3 . We made a sort of symmetric choice for the mapping in the subspace within C 2 , orthogonal to diagonal in the last line -but this is not canonical: what matters is just the mapping of this entire two-dimensional space onto its two-dimensional counterpart in C 3 .
In a similar way one can define morphisms in all other channels. In the rest of this subsection we present the tables of multiplicities, cohomologies and the superpolynomials in all the four channels for the trefoil knot diagram, in unreduced and reduced cases. 6.6.3 Unreduced trefoil in different channels bbw bww grads bbb bwb wbw www wbb wwb For initial vertex bbb we need to look at diagonal lines, decreasing from left to right. Contributing to cohomologies will be the lines with non-vanishing alternating sums. However, this time in the lower half of the table we have diagonals with the sums equal to 2. This defect of two can be distributed among the two possible columns in three different ways: 2 + 0, 1 + 1 and 0 + 2. The relevant choice is 1 + 1 and these are the cohomologies contributions a shown in boxes. This choice gives rise to the superpolynomial Similarly for initial vertex www we need to pick up diagonals, decreasing from right to left. Again there are different possible distributions of the defects, this time in the upper half of the table. The relevant choice is again 1 + 1, the corresponding cohomologies are double-boxed and the superpolynomial is For initial vertex bbw we need to rearrange the columns: bww bwb grads bbw wbw bbb wbb www wwb Diagonals are decreasing from left to right, location of non-vanishing cohomology is boxed, so that Similarly for initial vertex bww: bww bwb grads bbw wbw bbb wbb www wwb Diagonals are again decreasing from left to right, and 6. 6.4 Reduced trefoil in three different channels bbw bww grads bbb bwb wbw www wbb wwb For initial vertex bbb we need to look at diagonal lines, decreasing from left to right. Contributing to cohomologies will be the lines with non-vanishing alternating sums -remaining contributions are in boxes. Collecting all the three we get Similarly for initial vertex www we need to do the same with diagonals, decreasing from right to left. The corresponding cohomologies are double-boxed, and For initial vertex bbw we need to rearrange the columns: bww bwb grads bbw wbw bbb wbb www wwb Diagonals are decreasing from left to right, location of non-vanishing cohomology is boxed, so that Similarly for initial vertex bww: bbw wwb grads bww bwb wbb wbw bbb wbb Diagonals are again decreasing from left to right, and

k-folds
With above experience we are now ready to describe the entire series of 2-strand knots and links and reproduce the well known result of [35,36,37,38] from our version of Khovanov's construction.

Betti numbers for arbitrary 2-strand torus knots
According to [37], reduced polynomials for 2-strand knots are: what means that the quantum Betti numbers are reduced case : 1, 0, q 4 , q 2N +2 , q 8 , q 2N +6 , q 12 , q 2N +10 , . . . , q 4k , q 2N +4k−2 (107) Unreduced superpolynomials are less available in the literature, but from our above considerations it is clear that for the 2-strand knots one should just introduce factors [N ] for the zeroth Betti number and [N − 1] for all the rest and slightly modify the gradings: For q = 1 we get an extremely simple pattern: so that the sum rule (21) is nicely satisfied. In secs. 6.7.3 and 6.7.4 we demonstrate that these answers can be easily deduced from our construction.

Betti numbers for arbitrary 2-strand torus links
For links the structure of generic answers is more subtle. From [37] we know that links are associated with superseries (with all coefficients positive), rather than polynomials (we denote this quantity by underlined P ), and in reduced case In passing to Khovanov-Rozansky superpolynomial the underlined term should be eliminated (just erased) to convert the series into a finite polynomial (see [25] and remark after eq.(34) in [46]). Thus Khovanov-Rozansky polynomial, implied by [37] is where one finally substitutes a = q N .
For k = 1 this gives q N −1 1 + For q = 1 we get: in accordance with (21), because the 2-strand link has exactly two components. Now we proceed to the derivation of these results from our approach.

How this works. Reduced case
In fact it is sufficient just to redraw our pictures in appropriate way. Namely, put the lowest (in grading) constituents of all spaces C i in the first line, then the next -into the second, and so on. This makes the structure of morphisms absolutely transparent and cohomologies trivial to evaluate. Hopf: The first line has a single non-vanishing cohomology in the first term, the kernel contains one element of dimension H 0 = q 1−N . The fact that there is nothing else follows from 1 − 2 + 1 = 0. Nothing is mapped to the second line, it is pure cohomology H 2 . Thus Trefoil: The first line has a single non-vanishing cohomology in the first term, the kernel contains one element of dimension q 1−N . The fact that there is nothing else follows from 1 − 3 + 3 − 1 = 0. In the second line the situation is different: 3−2 = 1 = 0. This what we do, we split the first item in this line 3 = 2+1. Then 2−2 = 0 and there is no cohomologies in this reduced line, while the remnant gets mapped into the third line, providing a new cohomology -because where the map of the weight −1 has non-vanishing kernel and coimage: q · q 2−N and q 2 · q N −2 respectively. In result The balance in lines is now: and unbalanced cohomologies provide 5-foil: The balance in lines is now: and unbalanced cohomologies provide Generic case: Note that combinatorial factors in above tables are the products of C k n and C j−1 k−1 for the item at the crossing of the k-th column and j-th line.
The balances are: Note that morphisms and differentials are again decreasing the grading by −1. This time encycled are elements Note that all the morphisms are the same as they were between the corresponding spaces in secs. 6.8.1 and 6. 8

Reduced case: Another Orthogonal channel ••
This case is literally the same as the previous one, the differential are again made from morphisms, familiar secs. 6.8.1 and 6.8.2 -this time they are all instead of in sec. 6.8.3. Cohomologies and reduced superpolynomial are also the same 6. 8.5 Unreduced case.
Unreduced situation is described exactly in the same way. It is just necessary to add one more dimension of the size [N ], orthogonal to all constituents of d 1 and d 2 . This simply multiplies everything by [N ] and gives: The two pictures for the Seifert (••) and orthogonal (••) channels are shown in Fig.1.

Twist knots
This is the series that we already analyzed in sec. 5.7, now we promote our description of HOMFLY to superpolynomials. According to [42] and [28] reduced superpolynomial in the fundamental representation is We remind that for k = 0 and F 0 = 0 we get unknot, for k = 1 and F 1 = −A 2 -the trefoil 3 1 and for k = −1 and F −1 = 1 -the figure eight knot 4 1 . In fact, for k > 0 one should multiply the whole expression by −1 to make all the terms positive. With the exception of unknot and trefoil the superpolynomial contains negative powers of T -this is because the twisted knots are not represented by Seifert vertex in the hypercube: the corresponding vertices the knot diagram have different colors, n • = 0 and normalization factor α n• • ∼ T −n• provides negative powers of T . In the case of the twisted and 3-strand torus families we restrict ourselves just to the simplest example, which lies at the intersection of two families and can illustrate the both: that of the knot diagram from s. 5.6, which (for different colorings) is either the trefoil 3 1 or the figure eight know 4 1 or the unknot. The grading tables in the first two cases are: As usual, analysis of these tables does not predict the answers for Betti numbers unambiguously. The remaining discrete freedom is fixed by explicit construction of morphisms. These, in turn, are severely restricted by the requirement, that whenever possible (when they are mapping the same spaces in the same order) morphisms coincide for different colorings, i.e. for 3 1 , 4 1 and unknot represented by the same diagram D.
Since this time even in the reduced case pictorial representations of vector spaces are multidimensional (squares rather than strips, i.e. the power of N in dimension of the vector space is greater than one), in order to minimize the cohomology, already the first differential d 1 consists of morphisms, acting in different directions -like it already happened above for the double-eight representation of the unknot.
We hope that this kind of ideas, underlying the art of morphism-construction, is to some extent clarified by the previous examples and do not go into further details here. A unified analysis of the whole series of 3-strand torus knots and twisted knots, as well as more complicated examples, is clearly within reach and will be presented elsewhere. This is important also to demonstrate how things work in the case of knots, which are not "thin" -the first such example is the torus knot [3,4].
In the rest of this paper we briefly outline a conceptual approach to definition of morphisms in a systematic way, from the first principles. Again, we just formulate the ideas, leaving important details to further clarification.
7 Appendix: towards the theory of cut-and-join maps In [23] we explained that behind the morphisms of the Khovanov construction for N = 2 actually stand the cut-and-join operators [50], which nowadays play an increasing role in different branches of quantum field theory. Now we are going to explain -without going into too many details -that this is also true in our generalization from N = 2 to arbitrary N . It looks plausible, that a systematic presentation of our approach, together with all potentially interesting deviations, is best discussed from this perspective. However,in this paper we give just a brief survey, leaving the details for another presentation.
This section can be considered as alternative continuation of sec. 6.2. Since it is not targeted at concrete results, we allow more deviations from the main line in the simple examples -to demonstrate the additional possibilities, provided by the tensor-algebra approach. They all open potential new windows to various generalizations. In particular we attract attention to the freedom in the choice of morphisms, including their gradings -which can lead at least to significant technical simplifications.

An example of eight. Cut and join operations
Now we can insert one vertex in the knot diagram D, this means that there will be two in hypercube H(D). Topologically this is still an unknot. but now we have two different representations for it, differing by the choice of the color of the vertex in D and by the choice of initial vertex in H(D).
As we already know from s. 5 For taming the emerging world of factor-spaces we suggest the following procedure. Second, there are arrows at the edges -in general quite different from those on original hypercube (appearing, when initial coloring is chosen and pointing towards the corresponding initial vertex of H(D)). The arrows on the edgesH(D) describe embeddings and they do not obligatory go all in one direction.
Additional delicate point is that these are embeddings of graded vector spaces. To understand what they are, we need to recall that we want quantum dimensions to match -and the two possibilities are: This means that at the level of basises the embeddings are either e I −→ e I ⊗ e N ± e N ⊗ e I of degree q 1−N or e I −→ e I ⊗ e 1 ± e 1 ⊗ e N of degree q N −1 -choice of any other e K instead of e 1 or e N would give unappropriate . Following our general intention to preserve all the symmetries of the tensor algebra we avoid considering asymmetric embeddings like e I → e I ⊗ e N (in practice they do not give anything new). Antisymmetric embeddings have non-vanishing kernels (e N or e 1 respectively) and are also non-suitable for our purposes. Thus it remains to choose arbitrarily between the remaining two options. In what follows we postulate that embeddings in the primary hypercube are of degree q 1−N (then for N = 2 we get q −1 , familiar from [4] and [23]) and are explicitly given by It corresponds to the first (underlined) decomposition in (133). For an obvious reason we call ∇ the cut operation. Note that the grading 1 − N for N = 2 is exactly the standard −1. Now we need a complementary join operation ∆. There are different natural choices, to fix the freedom we ask 3 it to have the same (negative) grading as ∇: Cut operation has no kernel, but join operation has a huge one:, Similarly, coimage 4 of ∆ is empty, while CoIm(∇) = span e i ⊗ e j , e i ⊗ e N − e N ⊗ e i = span(ij, iN ) Here and below we assume that large Latin indices I, J run from 1 to N , while small ones, i, j -from 1 to N − 1. We also introduced a shortened notation for the basis elements in V ⊗2 . To simplify the formulas we omit brackets between dim and Ker in (138 and below. The corresponding dimensions are actually given by (133): 5

The main hypercube
Now we can return to the main hypercube. Resolution at its black vertex is just a pair of Seifert cycles, and we associate with this vertex the vector space V b = V • = V ⊗2 . Now comes the main point: with a "difference" of cycles at the white vertex we associate a factor-space V • = V ⊗2 /V . 3 Another distinguished choice would be an "inverse" to ∇, with Ker(∆) = CoIm(∇) and with grading N − 1, but it does not seem to lead to Khovanov-Rozansky homologies. 4 Throughout this paper we understand "cokernel" and "coimage" as complements of the kernel and image in the initial and target spaces respectively. In general these are factor-spaces, but when basises are explicitly specified, thee can actually be considered as well defined orthogonal complements. 5 In reduced case everything looks even simpler: Here E with g(E) = 0 is the single basis element in E = C, and we denote it by the same letter as the space. Now, the two decreasing morphisms ξ and η look as follows: vertical lines are various spaces at the vertices ofH(O • O) and H(O • O) and arrows are identical (or nullifying) maps on their subspaces. Clearly, Ker(ξ) = Im(∇) is a result of embedding of U • = V into U • = V ⊗2 , but it differs from V itself in a shift of grading. The same is true for CoIm(η) -but the grading shift in this case is different. q From this picture it is clear that The two non-vanishing spaces,shown by thick line in the picture, are CoKer(∆) = span(e 1 ⊗ e I + e I ⊗ e 1 ) dim q CoKer(∆) = q N −1 [N ] As to CoKer(∆), it is "similar" to U • , but is am absolutely different subspace in V • = U • , in particular with a very different grading. 7.1.3 The choice of morphisms. "Gauge" invariance.
In this picture the vertical "coordinate" is actually the grading degree. Dots symbolize the elements (subspaces) with grading zero. The average grading of the space V • is m, which is an arbitrary integer: one can move V • arbitrarily along the vertical line -this reflects the absence of canonical representative in the equivalence class of vector subspaces, which the factor-space always is. In graded case the freedom is substantially restricted, still remains. Moving V • along the vertical line, one changes the gradings of morphisms ξ and η, but two things remain intact: and cohomologies of ξ and η in (140). This is somewhat similar to gauge invariance. As usual, Euler characteristic can be calculated in two ways -via dimensions of vector spaces V and via cohomologies: Note that the quantum dimension dim q V • = q m [N ][N − 1] and the gradings g(ξ) = m − 1, g(η) = 1 − m depend on the shift m.
The same remains true for arbitrary knot/link diagrams D. However, often it is more convenient to rely upon concrete (in no way canonical) choice of the representative for the factor-space. There are actually three technically distinguished choices: g(ξ) = g(η) = −1 or g(ξ) = 0 or g(η) = 0. The first case is more symmetric and it makes smoothconnection with the standard construction at N = 2 (where no factor-spaces occur in explicit way). The other two choices can be natural, if we look at particular coloring D c (fix the initial vertex of the hypercube): then only one of the two morphisms matter, and it is technically reasonable to simplify it as much as possible. Choosing grading degree zero for this morphism allows to make it simply an identity map between its cokernel and coimage -what makes calculations as simple as only possible. In other channels (other initial vertices) one can make another choice, making identical another relevant morphism. If we take this road, this actually means that we make different choices of the space V ⊗2 /V (different representatives of the class) in different channels, i.e. modify slightly the original definition of the main hypercube. As we just explained, this has technical advantages. At the same time, cohomologies of the complex K(D), their dimensions and thus the Khovanov-Rozansky polynomials do not feel the difference, if appropriately defined -as in (140). In more technical definitions appropriate adjustements will be needed: of the parameters in the generating function (it will be q g(ξ) T ) and overall coefficient.
We can illustrate the difference between "symmetric" and "identity" choices already now.
In this case we just take CoIm(∇) to represent V ⊗2 /V = span(ij, iN ) of dimension q[N ][N −1], i.e. by definition ξ is just an identity map on CoIm(∇), so that g(ξ) = 0. At the same time η maps it one-to-one onto Ker(∆) and has g(η) = −2: Clearly, i.e. just the same as in symmetric case above -in full accordance with (140).

A c -dependent gauge choice and general procedure
Clearly, identity maps of grading zero are much simpler to deal with. Moreover, using them we make a conceptual simplification: instead of arbitrary factor-spaces we consider canonically defined Ker(∆) and CoIm(∇). The price to pay for this is to allow the vector spaces at the hypercube vertices to depend on initial vertex, i.e. on the coloring c of vertices in the knot/link diagram D c . For each particular c we deal either with ξ or with η, but not with the both together. Therefore, for each particular c we can choose the spaces V so that the relevant maps are of the grading degree zero. For another coloring we shift the spaces so that the new maps have vanishing degree. In our current example, we have just two choices of c: with initial vertex black (left picture) and white (right picture): Finally. the morphisms can be read from (145) and (147): Sometime we will simply write ∇ and ∆ instead of ξ ∇ and η ∆ .

Associated complex and unreduced superpolynomials
In this particular case of a one-dimensional hypercube, the last step -building a complex from a commutative quiver -is just trivial ( Therefore the cohomologies of these complexes and their Poincare polynomials can be just read from (140). Khovanov-Rozansky superpolynomial is obtained from Poincare polynomial by adding a simple overall factor: Since our morphisms along the edges of H(D) have grading 0, the weights in the sum in Poincare polynomial are powers of T , not qT . 6 Actually, the value of this factor can be obtained from the requirement that superpolynomials for the eight are the same as for the circle -and then used in calculation for all other examples. Finally, the superpolynomials for the eight for initial vertex black an white are respectively and 6 Note that in sec.5 we tried to keep close to sec.4, thus the factors were different and the weight was made out of qT . It is an interesting question, if one can construct some other set of morphisms in H(D), with non-vanishing grading, to match those formulas. However, the morphisms of grading zero seem extremely natural in our construction. To restore matching with sec. and coincidence between the two polynomials is now just explicit term-by term. The price to pay is explicit appearance of q-factors in the space dimensions -which would be un-understandable in sec.5, but gets clear now, when the spaces and morphisms are explicitly defined.

Reduced superpolynomials
As explained in sec. 6.2.2, reduced superpolynomial is obtained by the same construction, only one vector space per vertex of the hypercube, associated with a cycle, passing through a marked edge in D, should be reduced from N -dimensional V to 1-dimensional E. Clearly, in our construction this should be done at the level of primary hypercube, which is now (for the eight) Since we want the cut-operation ∇ to always have grading q 1−N , the choice is predefined, see s. 7.1.1: Here E with g(E) = 0 is the single basis element in E = C, and we denote it by the same letter as the space. At two vertices of the main hypercube we now have vector spaces V ⊗ E = V and V ⊗ E/E = V /E with the maps (morphisms) and with This gives the proper reduced superpolynomials

Double eight. Combinations of cut and join
In the next example the number of vertices in D is 2, thus hypercube H(D) is a 2-dimensional square (or rhombus). In fact there are two different D with two vertices: double eight, consisting of three circles, which is the unknot for any coloring, and two circle intersecting at two points -depending on coloring this is either a Hopf link ar two disconnected unknots. We begin in this section from the double eight example.

Primary hypercube
The starting point of our construction is the primary hypercube: with embedding maps of degree 1 − N are: In the first case all the arrows in the hypercube are opposite to those in the primary hypercube, while in the second case they coincide with those. Accordingly all the arrows in the first case are of the ∇ (cut) type, and of the ∆ (join) in the second (note that the maps in the two pictures are actually ξ and η, while ∇ and ∆ play here the role of labels, marking the type of the spaces -factor or sub -and of the morphisms): The choice of spaces V •• and V •• follows our previous example for a single eight in sec. 7 However, they are combined in different ways: When two ∇ arrows are entering the vertex ••, we associate with this vertex a factor space over a union of these two spaces: U •• + U •• = span U •• , U •• which we denote by an ordinary plus sign in what follows. When two ∆ arrows are quitting the vertex ••, we associate with this vertex a subspace, complementing the intersection of the two: From now our consideration of the two cases splits for a while, only to merge again at the end of this section.
The following picture is for the case of c = •• . If one wants all the spaces in this picture to be represented by segments (rather than consists of two pieces, cometime) it is enough to imagine that the space U •• is a circle, i.e. the points a and e = a coincide, and segment yx is a complement of xy.
Since we choose our morphisms ξ i to be of degree zero, all the factor-spaces can be de facto identified with the subsets of V •• , where they act as identities, so that our quiver is obviously Abelian: With this Abelian quiver one naturally associates a complex were the two differentials are and d 1 d 0 = 0 as a corollary of (162). Superpolynomial is just a Poincare polynomial of this complex, multiplied by additional factor (151): It is clear from the picture that while Im(d 0 ) = Ker(d 1 ) and CoIm(d 1 ) = ∅, so that is indeed equal to the superpolynomial for the unknot. The same is true in the reduced case, where the only change is that the intersection (161) reduces to just a single element e N ⊗ e N , which has quantum dimension q 2(1−N ) , so that the reduced superpolynomial is Remarkably in the case of c = •• we can draw just the same picture, only upside-down. What is different is just interpretation of different segments: one changes ∇ for ∆, coimages for kernels and factor-spaces for 1 It is clear from the picture that Ker(d •• 1 ) = ∅ and Im(d Note, that in our pictures d e can look similar to de, but in fact they have different gradings -different by a change q −→ q −1 . In the reduced case, where the only change is that the intersection of cokernels reduces to just a single element e 1 ⊗ e 1 , which has quantum dimension q 2(1−N ) , so that the reduced superpolynomial is For two other initial vertices, •• and ••, the situation is a little more tricky, because where two merging edges at the vertices of the hypercube are of different types -one ∇ and one ∆. In order to handle such configurations we need still one more reformulation of our approach. r r r r

A slight reformulation
We now draw the same pictures in a slightly different form From (160) we know explicit expressions for the maps: In fact, the non-trivial cohomology in this case is given by = span e I ⊗ e N ⊗ e N + e N ⊗ e I ⊗ e N + e N ⊗ e N ⊗ e I In this particular case the two intersecting consequtive images are actually the same.
To avoid possible confusion, we note that in our picture a part of ∇ 1 looks like a gradation-increasing map -this is an artefact of the drawing, actually all ∇ maps are of degree 1 − N . The same applies to a part of ∆ 1 in the next picture.
The join operations act as follows: The non-trivial cohomology in this case is

The main hypercube and the superpolynomials with initial vertices •• and ••
Now the situation is a little more tricky, because we have vertices, where two edges of different type -∇ and ∆ -merge together.
The choice of spaces V wb and V bw follows our previous example for a single eight in sec. 7 , which we will often denote simply by U wb + U bw . Existence of U ww implies that the images of these two subspaces, within U bb can intersect.
In other words, the embedding pattern is as follows: Here all spaces U v and V v are shown as embedded into the largest one, V bb = U bb (in this picture the factor-space The only comment can be needed in the case of the last space V bb : it is a complement in V ⊗3 to V ⊗2 + V ⊗2 ≡ span(V ⊗2 , V ⊗2 ) = span (e I ⊗ e N + e N ⊗ e I ) ⊗ e K , e I ⊗ (e N ⊗ e K + e K ⊗ e N ) , and these two sets have a non-trivial intersection, when embedded into U bb Underlined product is what we took for dimension of this space in (26), while q 2 appeared there as additional weight in the definition of the HOMFLY polynomial. Similarly, dimensions include the factor q.

Morphisms and superpolynomials
Pictorially the structure of hypercube and morphisms is: In this picture the bb vertex is at the center, wb and bw are to the right and to the left of it respectively, and the ww vertex is shown twice -in the very right and the very left column.
From this picture we immediately see what happens when initial vertex is, say, bb. Then Only the third line here requires a comment: from the picture it can seem that there is no kernel at all, but in fact, because the ww is shown twice, there are two arrows directed to it, moreover the left arrow acts with the minus sign -therefore the kernel is non-vanishing and given by above formula. Thus unreduced superpolynomial As to reduced polynomial, note, that this time there are two a priori inequivalent choices of the marked edge in D: on external circle and on internal circles of double eight. However, in both cases the spaces at the vertices of auxiliary hypercube are reduced the same way: to Actually such are the spaces, when a middle edge in D is marked. If it was instead an edge with both ends at bb, we would rather get U bw = E ⊗V -however it is again the same as V , and in both cases Emb(U bw )∩Emb(U wb ) = Emb(E) = span(e N ⊗ e N ) with dim q Emb(U bw ) ∩ Emb(U wb ) = q 2−2N reduced case (188) so that the reduced superpolynomial is Similarly, for the other choices of initial vertex: and Note that -as clear from the above pictures -the only non-vanishing contribution to all these formulas comes from (182) and (188), this is why we put them in boxes.

The general procedure for the choice of the vector spaces
We end this preliminary presentation of the cut-and-join formalism behind our version of KR calculus by formulating the general rule, for the choice of vector spaces at the vertices of the hypercube H(D). This choice is straightforwardly dictated by the structure of the primary hypercubeH(D). In the sub-cube h v there will be drain vertices -where all arrows enter and no one exits. Since arrows describe embeddings, one can factor a vector space at the drain vertex over a span of all embedded spaces at the origins of the entering edges. Finally we associate with a hypercube vertex v ∈ H(D) (in original hypercube) a sum of these factor-spaces over all drain-vertices d, belonging to the corresponding sub-cube h v : The space in denominator of the factor is spanned by a combination U w at the vertices w in h v , which are preimages of the given drain point d, i.e. by definition of the drain point all of them are embedded into U d , but can intersect -and after that we take a direct sum of such factor-spaces over all drain points in h v . What is important, the drain points and their preimages are taken not from entire hypercube, but from the v-dependent sub-cube h v . Note that all vector spaces U v at the vertices ofH(D) are just tensor powers of V = C N . Eq.(194) looks like a terribly complicated formula, but, hopefully, after working through several examples from sec.6, its simple meaning gets perfectly clear.
The r.h.s. of (194) is somewhat symbolic, because it is important how the vector spaces in "denominators" are embedded in those in the "numerators", i.e. how the factor-spaces are actually defined. This is, however, straightforwardly dictated by the embeddings of spaces U v along the edges of the primary hypercube, i.e. by the cut operation (134). After that the cut and join operations (134) and (135) define what are all the morphisms in all directions along the edges of the main hypercube. They form a commuting set -an Abelian quiver. Therefore, once initial vertex c 0 is chosen in the main hypercube, one can always construct associated complex K(D c0 ). After appropriate normalization its Poincare polynomial coincides with Khovanov-Rozansky polynomial, obtained by a very different and far more complicated matrix-factorization technique.

Conclusion
In this paper we suggested an alternative construction of Khovanov-Rozansky superpolynomials for arbitrary knots and links and for arbitrary gauge group GL(N ). It is completely different from the original matrixfactorization construction of [7] and our calculations have nothing in common with those of [19] -except for the answers. Moreover, in our way we get the answers for all values of N at once. Also calculations are extremely simple and easily computerizeable -probably even the programs, used for the calculations of Jones superpolynomials (i.e. for N = 2), can be easily modified and used for arbitrary N . In the paper the simplest examples are done "by hands", and these include the big part of the list of [19], obtained by extremely tedious computer calculus at particular N . Moreover, we explained how the entire series of 2-strand k-folds can be handled. Extensions to other series, beginning from twist and 3-strand torus knots would be a natural next step to do.
The paper concentrates on the ideas, and does not present the story as a systematic algorithmic approach -the ways to do this are outlined but nor developed to the very end. Accordingly, no general proof is given of the Redemeister invariance. These issues, as well as the relation to the matrix-factorization formalism, to Hecke-algebra [39,51] and to refined-Chern-Simons [52]- [56] approaches will be discussed elsewhere.
Of more importance, however, would be practical calculations, making the list of the Khovanov-Rozansky polynomials as rich as that of HOMFLY. Moreover, this approach can probably be more than competitive in HOMFLY calculations themselves, like it already is for N = 2, see s.4 in [28]. The next step should be extension from fundamental to antisymmetric, symmetric and arbitrary representations -where already for HOMFLY the standard R-matrix approach [41,32,43,45] gets extremely tedious and too few results are available, what slows down the progress in the field. It looks like these extensions can also be found, by application of the same tensor-algebra vision of [31] which led to the important success, reported in the present paper.