Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial

Transcription

1 Virtual crossings, convolutions and a categorification of the SO(2N Kauffman olynomial M. Khovanov 1 Deartment of Mathematics, Columbia University 2990 Broadway 509 Mathematics Building New York, NY address: khovanov@math.columbia.edu L. Rozansky 1 Deartment of Mathematics, University of North Carolina CB #3250, Phillis Hall Chael Hill, NC address: rozansky@math.unc.edu Abstract We suggest a categorification rocedure for the SO(2N one-variable secialization of the two-variable Kauffman olynomial. The construction has many similarities with the HOMFLY-PT categorification: a lanar grah formula for the olynomial is converted into a comlex of graded vector saces, each of them being the homology of a Z 2 -graded differential vector sace associated to a grah and constructed using matrix factorizations. This time, however, the elementary matrix factorizations are not Koszul; instead, they are convolutions of chain comlexes of Koszul matrix factorizations. We rove that the homotoy class of the resulting comlex associated to a diagram of a link is invariant under the first two Reidemeister moves and conjecture its invariance under the third move. 1 This work was suorted by NSF Grants DMS and DMS

2 1 Contents 1. Introduction SO(2N + 2 Kauffman olynomial and graded comlexes Closed grahs and the alternating sum formula The outline of the categorification A categorification comlex of a virtual link Acknowledgements Matrix factorizations and grahs A category of matrix factorizations Koszul matrix factorizations Two theorems Grahs, tangles and associated matrix factorizations Arc matrix factorizations Basic olynomial Jacobi algebra The 1-arc Koszul matrix factorization and the unknot sace arc Koszul matrix factorizations Saddle morhisms A roer saddle morhism Comositions of saddle morhisms Semi-closed and closed saddle morhisms Multi-dimensional Postnikov systems and their convolutions A multi-dimensional Postnikov system Building a Postnikov system from the bottom u Corner subsystems and factorsystems Graded Postnikov systems Examles of Postnikov systems and convolutions Categorifiction of the 4-vertex, grahs and tangles A convolution of the chain of saddle morhisms 47

3 The 4-vertex matrix factorization and the crossing comlex A categorification comlex The Reidemeister moves Excision of a contractible cone First Reidemeister move Second Reidemeister move Virtual and semi-virtual Reidemeister moves 68 References 69 Aendix A. Virtual crossings, convolutions and the categorification of the HOMFLY-PT olynomial and of its SU(N secialization 70 A.1. The SU(N case 70 A.2. The 2-variable HOMFLY-PT olynomial case 78 Aendix B. The isomorhism between the SU(2 SU(2 and SO(4 categorifications 81 B.1. The isomorhism theorem 81 B.2. The SU(2 categorification for unoriented link diagrams 81 B.3. Proof of the isomorhism of comlexes Introduction 1.1. SO(2N +2 Kauffman olynomial and graded comlexes. The SO(2N +2 Kauffman olynomial [4] P L (q Z[q ±1 ] is an invariant of an unoriented framed link L S 3, which satisfies the skein relation [ ] [ ] ( [ ] [ ] P P = (q q 1 P P, (1.1 the change of framing relation [ ] [ ] P = q 2N+1 P (1.2

4 3 (throughout the aer we assume that links are endowed with the blackboard framing, and the multilicativity roerty P L1 L 2 (q = P L1 (qp L2 (q, (1.3 where L 1 L 2 is the disjoint union of the links L 1 and L 2. The conditions (1.1 (1.3 determine P L (q uniquely [4], [6]. In articular, the Kauffman olynomial of the unknot is equal to P unkn (q = q2n+1 q 2N+1 q q (1.4 We will describe a conjectural categorification of the SO(2N + 2 Kauffman olynomial. Namely, to a link diagram L we associate a chain comlex C (L of Z 2 Z-graded Q-vector saces C n (L C n+1 (L, C n (L = n,j C{i} (L. (1.5 i Z j Z 2 We rove that, as an object in the homotoy category of comlexes of Z 2 Z-graded vector saces, the comlex C (L behaves nicely under the first two Reidemeister moves: Theorem 1.1. The comlex C (L is homotoy invariant u to degree shifts under the first Reidemeister move: C [ ] C [ ] { 2N 1} 1 [ 1], (1.6 Here { },, and { } denote the shifts in the homological degree of the comlex (1.5, in the Z 2 -degree and in the Z-degree resectively. Theorem 1.2. The comlex C (L is homotoy invariant under the second Reidemeister move: C C. (1.7 Further, we conjecture that Conjecture 1.3. The comlex C (L is homotoy invariant under the third Reidemeister move: C C. (1.8

5 4 Conjecture 1.4. The whole comlex C (L has a homogeneous Z 2 -degree where n L is the number of crossings in the diagram L. deg Z2 C (L = n L (mod 2, (1.9 We will show that Conjecture 1.3 imlies that the graded Euler characteristic of the comlex C (L ( χ q C (L = equals the Kauffman olynomial: ( 1 i,n Z j Z 2 j+n q i dim C n,j {i}(l (1.10 χ q ( C (L = P L (q. (1.11 Our aroach to the construction of the categorification comlex C (L is similar to that of [8]: it follows the alternating sum formula for the olynomial P L (q, which exresses it in terms of olynomial invariants of lanar grahs. The categorification of elementary oen grahs is rovided by matrix factorizations, and this time the basic olynomial is W(x,y = xy 2 + x 2N+1, as suggested by Gukov and Walcher [2] Closed grahs and the alternating sum formula. Kauffman and Vogel [6] extended the Kauffman olynomial from links to knotted 4-valent grahs. They defined the invariant of grahs by resenting the 4-vertex as a linear combination of crossings and their resolutions: [ ] [ ] [ ] [ ] P = P + qp + q 1 P [ ] [ ] = P + qp + q 1 P [ ]. (1.12 The second equality follows from the skein relation (1.1 and exresses the 90 rotational symmetry of the 4-vertex. The formula (1.12 resents the Kauffman olynomial of a knotted grah as a linear combination of Kauffman olynomials of links constructed by resolving each 4-vertex in three ossible ways, thus reducing the comutation of the Kauffman olynomial of a grah to that of links. The relation (1.12 can also be layed backwards, that is, a crossing can be resented as a linear combination [ ] [ ] [ ] [ ] P = qp P + q 1 P. (1.13

6 5 In this form, it allows us to exress the Kauffman olynomial of a link in terms of the olynomials of lanar grah diagrams. Namely, let L be a link diagram with n L enumerated crossings. To a multi-index r = (r 1,...,r nl, r i { 1, 0, 1}, we associate a lanar grah diagram Γ r constructed by resolving all crossings of L: the i-th crossing is resolved in r i -th way. Then, according to eq.(1.13, P L (q = ( 1 n L ( 1 r q r P Γr (q, (1.14 where r = n L i=1 r i. We call 4-valent grahs Γ r closed. r 1.3. The outline of the categorification. Similar to the HOMFLY-PT olynomial case, the categorification of the Kauffman olynomial is based on turning the alternating sign sum (1.14 into a comlex of graded vector saces. To each closed grah Γ r we associate a Z 2 Z-graded vector sace C (Γ r (in fact, we conjecture that deg Z2 C (Γ r = 0; the aearance of the Z 2 degree n L in the formula (1.9 is due to the Z 2 degree shift in eq.(1.16. We assemble all saces C (Γ r into an n L -comlex by lacing them according to their multi-indices r (interreted as n L -dimensional coordinates at the nodes of a Z n L -lattice and defining aroriate differential mas between adjacent saces. Namely, if three grahs Γ, Γ, Γ result from the same resolution of all crossings of L, excet for an i-th crossing, where they differ according to their indices, then we construct the linear mas forming the chain comlex C ( Γ {1} χ ( in,i χout,i ( C Γ C Γ { 1}. (1.15 The comlexes (1.15 reresent the action of n L mutually anti-commuting differentials, each differential being related to a resolution of a articular crossing of L. The categorification comlex C (L results from collasing the n L -comlex into a single comlex by taking the sum of individual differentials and shifting the Z 2 -degree of the whole comlex by n L units. The homological degree of each sace C (Γ r is set to be equal to r. For examle, the comlex of the Hof link is resented in Fig. 1. There the mas χ,u and χ,d are related to the uer and lower crossing of the Hof link diagram. In order to construct the saces C (Γ r and the morhisms between them, we break the link diagram L and the corresonding closed grahs Γ r into simle ieces by cutting across each edge of the diagram L. The diagram L slits into elementary tangles, while the closed grahs Γ r slit into elementary oen grahs, and. To each elementary oen grah γ we associate a matrix factorization ˆγ, and to an elementary tangle we

7 6 C = C C {2} χ in,d {1} C χ out,d χ in,u χ in,u χ in,u C C {1} C χ in,d χ out,d χ out,u C χ in,d χout,u C { 1} { 1} χout,u C, { 2} χ out,d Figure 1. The categorification comlex of the Hof link associate a comlex of matrix factorizations ( χ in = {1} χ out { 1} 1 (1.16 with a secial choice of morhisms χ in and χ out, which are local versions of the morhisms (1.15. The homological degree of the middle matrix factorization in this comlex is set to zero. If a closed grah Γ r consists of n L elementary oen grahs γ i, then we set C (Γ r = H MF (ˆΓ r, ˆΓr = C (L = H MF (ˆL, n L i=1 i=1 ˆγ i, (1.17 n L ˆL =, (1.18 i and H MF is the matrix factorization homology (its definition is given in subsection 2.1; the recise definition of the tensor roduct aearing in the formulas for ˆΓ r and ˆL will be given in subsection 2.4. As a tensor roduct of comlexes (1.16, C (L will have an exected structure of the total comlex of the n L -dimensional cube-like comlex. The first challenge of the categorification of the Kauffman olynomial is to choose the right matrix factorization and morhisms χ in,χ out. It turns out that in contrast to the

8 7 SU(N and HOMFLY-PT cases we can not resent as a Koszul matrix factorization (its rank is not a ower of 2. Instead, matrix factorizations. is realized as a convolution of a comlex of Koszul First of all, we introduce a new tye of 4-vertex for grahs: a virtual crossing. Its matrix factorization is just a tensor roduct of two 1-arc matrix factorizations (similarly to and, as if the segments of did not cross. Then we define two secial morhisms F G, (1.19 which we call saddle morhisms, and we rove that GF 0. (1.20 The category of matrix factorizations is triangulated, which means that to a morhism A f B between two matrix factorizations one can associate a matrix factorization Cone MF (f called the cone of f. The cone comes with a air of natural morhisms which form two sides of the exact triangle of f. B Cone MF (f A 1, (1.21 A similar construction can be alied to a chain of two morhisms A f B g C, such that gf 0. The resulting matrix factorization is called a convolution and we denote it by a frame box around the chain: A f B g C (note that here we omitted the secondary homomorhism in order to simlify the diagrams; the role of secondary morhisms will be exlained in Section 5. The module of the convolution is a sum of modules A B C. The convolution comes with two natural morhisms, which form a chain comlex of matrix factorizations: ( 00 C id A f B g C (id 0 0 A. (1.22 This allows us to define as a convolution of the chain (1.19 = { 1} F 1 G {1} (1.23

9 8 and use the natural morhisms of (1.22 as χ in and χ out, so that the definition (1.16 of the categorification of a crossing becomes = ( 00 {1} id { 1} F 1 G {1} (id 0 0 { 1} 1. (1.24 In view of the exlicit form (1.22 of the morhisms χ in and χ out, we may deict the diagram (1.24 informally as id { 1} { 1} F = 1 1 (1.25 G id {1} {1} If we substitute the definition (1.23 into the diagram of Fig. 1, then the comlex for the Hof link takes the form deicted in Fig. 2. We omitted there degree shifts and secondary homomorhisms and used an abbreviated notation ( = H MF (, (1.26 where is a chain of matrix factorization morhisms A categorification comlex of a virtual link. Since we have introduced the matrix factorization for a virtual crossing, it is natural to extend the definition (1.18 of the categorification comlex of a link to virtual links. Virtual links were invented by L. Kauffman [5] and their toological meaning was clarified by G. Kuerberg [10] (for the details see V. Manturov s book [11]. From the combinatorial oint of view, they are equivalence classes of virtual link diagrams. A virtual link diagram is a 4-valent lanar grah with two tyes of 4-valent vertices: an ordinary crossing and a virtual crossing. Two diagrams are equivalent (and thus resent the same virtual link, if one can be transformed into another by a sequence of secial moves: ordinary Reidemeister moves, virtual Reidemeister moves and semi-virtual Reidemeister moves.

10 9 χ in,u F u G u χ out,u χ in,d χ in,d χ in,d F u G u F d χ in,u F d F u F d G u F d χ out,u F d G d G d F u G d G u G d G d χ out,d χ in,u F u χ out,d G u χ out,u χ out,d Figure 2. Detailed structure of the Hof link categorification comlex The virtual Reidemeister moves are the ordinary Reidemeister moves, in which all crossings are relaced by virtual crossings. A semi-virtual Reidemeister move is a third Reidemeister move, in which two crossings are relaced by virtual crossings:. (1.27

11 10 Let L be a virtual link diagram with n L 4-vertices. We break it into ieces γ i (1 i n L by cutting across its edges. We call these ieces elementary tangles. The elementary tangles γ i coming from the breaku of L are either crossings or virtual crossings. We define the categorification comlex C (L of a virtual link diagram L similar to eq.(1.18: C (L = H MF (ˆL, n L ˆL = ˆγ i, (1.28 i=1 Theorem 1.5. If two virtual link diagrams L and L are related by a virtual or a semi-virtual Reidemeister move, then their comlexes C (L and C (L are homotoy equivalent. We will rove this theorem in subsection Acknowledgements. We want to thank Yasuyoshi Yonezawa for helful suggestion regarding the manuscrit and Vassily Manturov for a discussion of roerties of virtual links. L.R. is indebted to his wife Ruth for her invaluable suort. This work was suorted by NSF Grants DMS and DMS Matrix factorizations and grahs 2.1. A category of matrix factorizations. Let us recall the basic definitions and notations related to matrix factorizations. For a olynomial W in a olynomial ring R, we define a (homotoy category of matrix factorizations MF R,W (or simly MF W. Its objects are Z 2 - graded free R-modules M = M 0 M 1 equied with the twisted differential D End R (M such that deg Z2 D = 1, D 2 = W id. (2.1 Thus D slits into a sum of two homomorhisms D = P + Q If W 0, then the conditions (2.1 imly M 1 P M 0 Q M 1, PQ = QP = W id. (2.2 If W = 0, however, then the relation (2.3 no longer holds. rankm 0 = rankm 1. (2.3 Sometimes, when R = Q[x], x = x 1,...,x n, we may use a notation M x (for a subset {x } {x} instead of simly M. The advantage is that we can then denote by M y the matrix factorization obtained from M x = M by renaming some of the variables x, namely x, into y.

12 11 The twisted differential D generates a differential d (d 2 = 0 acting on R-homomorhisms F Hom R (M,M between the underlying R-modules M and M by the formula where we use the notation df = [D M,M,F] s, (2.4 D M,M = D M + D M, (2.5 while [, ] s is the Z 2 -graded commutator. In our conventions a comosition of two homomorhisms is zero, unless the target of the first one matches the domain of the second one. Then we define the Z 2 -graded extensions as the homology of d: Ext (M,M = kerd/ im d, Ext (M,M = Ext 0 (M,M Ext 1 (M,M, (2.6 and the morhisms between M and M in the category MF R,W are defined as Hom MF (M,M = Ext 0 (M,M. (2.7 If P = id, Q = W id or P = W id, Q = id, then the identity endomorhism in End(M becomes a trivial element in Ext 0 (M,M, so the corresonding matrix factorization is isomorhic to the zero object in MF W. We call the latter matrix factorizations contractible. If W = 0, then D 2 = 0 and we can define the homology of the matrix factorization M as If in addition to that R = Q, then H MF(M = kerd/ im D. (2.8 M H MF(M (2.9 as objects in the category MF Q,0 of Z 2 -graded differential vector saces. The translation functor 1 turns the matrix factorization (2.2 into Q P M 1 = (M 1 M 0 M 1, (2.10 the double translation 2 acts as the identity and Hom MF (M,M 1 = Ext 1 (M,M. (2.11 For M Ob (MF R,W and M Ob (MF R,W we define a tensor roduct M M Ob (MF R R,W+W (2.12 as a matrix factorization, whose module is the tensor roduct of modules M M and whose twisted differential is the graded sum of twisted differentials D id + ( 1 deg Z 2 id D. (2.13

13 12 Similarly, for M Ob (MF R,W, M Ob (MF R,W we define a tensor roduct M R M Ob (MF R,W+W (2.14 as a matrix factorization, whose module is the tensor roduct of modules M R M and whose twisted differential is (2.13. The conjugate of a matrix factorization (2.2 is the matrix factorization M of W: M 1 Q M 0 P M 1 (2.15 It satisfies the roerty that for any matrix factorization M of W, Ext (M,M = H MF(M R M (2.16 (note that according to (2.14 M R M Ob (MF R,0, so the r.h.s. is well-defined. For a homomorhism R h R between two olynomial algebras there is a functor MF ĥ R,W MF R,h(W (2.17 which takes a matrix factorization M to M R R. There is a homomorhism R ˆ End MF (M, (2.18 which turns an element r R into a multilication by r: ˆr(v = rv, v M. (2.19 If R = Q[x], x = (x 1,...,x m, then the homomorhism (2.18 factors through the Jacobi algebra J W = Q[x]/( x1 W,..., xm W, (2.20 that is, the homomorhism (2.18 is a comosition of homomorhisms ˆ Q[x] J W End MF (M. (2.21 Finally, if R is a Z-graded ring and W has an even homogeneous Z-degree deg Z W = 2N, then one can define a category of Z-graded matrix factorizations of W. Its objects are matrix factorizations with (Z 2 Z-graded modules M, whose twisted differentials have a homogeneous Z-degree deg Z D = N.

14 Koszul matrix factorizations. Many of our constructions are based uon Koszul matrix factorizations. For two olynomials,q R the Koszul matrix factorization K(;q of W = q is a free Z 2 -graded R-module R 0 R 1 of rank (1, 1 with the twisted differential D, whose action is It is easy to verify that R 1 that is, both ˆ and ˆq are homotoic to 0. R 0 q R 1. (2.22 ˆ, ˆq 0, (2.23 If both olynomials, q have homogeneous (although, maybe, unequal Z-degrees, then the Koszul matrix factorization (2.22 can be made Z-graded by shifting the degree of R 1 by k,q = 1 2 (deg Z deg Z q. (2.24 Hence when dealing with graded Koszul matrix factorizations we use the notation K(;q = ( R 1 {k,q } R 0 q R 1 {k,q }, (2.25 where { } denotes the Z-grading shift. K(;q {k} denotes the Koszul matrix factorization in which the Z-grading of both modules R 0, R 1 is shifted by extra k units relative to that of (2.25. Obviously, K(;q {k} 1 = K( q; {k,q + k}, ( K(;q {k} = K(q; { k}. (2.26 For two columns = 1. n, q = q 1. q n, i,q i R, i = 1,...,n, (2.27 we define the Koszul matrix factorization K(;q of the olynomial n i=1 iq i as the tensor roduct of the elementary ones: n K(;q = K( i ;q i. (2.28 i=1 The case n = 2 lays an imortant role in our comutations, so in order to set the basis notations, we resent it exlicitly. Thus consider a Koszul matrix factorization ( 1, q 1 1 q 1 K = (R 1 R0 R1 (R 2 1 R q 2 2, q 0 R 1 ( We choose the generators of rank 2 submodules of Z 2 -degree 0 and 1 according to the slitting R 2 0 = R 00 R 11, R 2 1 = R 01 R 10, where R ij = R i R j. (2.30

15 14 Then the Koszul matrix factorization (2.29 has the form R 2 1 P R 2 0 Q R 2 1 (2.31 where P = ( ( 2 1 q 2 1, Q =. (2.32 q 1 q 2 q 1 2 In fact, the columns of ring elements (2.27 should be considered as elements of R n and (R n : R n, q (R n, W = q. (2.33 The imlication is that a articular choice of olynomials i and q i reresenting and q deends on the choice of free generators of R n. A change of generators modifies i s and q i s, but the Koszul matrix factorization K(;q remains the same. In articular, it is invariant under the following row transformations, acting on i-th and j-th rows of the columns (2.27: ( i j, ( q i q j ( ( [i,j] λ i q i λq j,, λ R. (2.34 j + λ i q j A conjugation of this transformation by the translation functor 1 (which may act by switching j and q j yields another useful equivalence: ( ( ( ( i q i [i,j] λ i q i λ j,,, λ R. (2.35 j q j q j + λ i j For an invertible element λ R there is another equivalence transformation acting on the i-th row of and q: ( i,q i [i] λ (λ i,λ 1 q i ( Two theorems. The following two theorems resent us with convenient technical tools for establishing equivalences between Koszul matrix factorizations. Theorem 2.1. If 1,..., n R form a regular sequence and n n i q i = i q i, (2.37 then the following two Koszul matrix factorizations are isomorhic: i=1 i=1 K(;q = K(;q (2.38

16 15 Proof. We rove the theorem by induction over n. If n = 1, then the claim is obvious. Suose that the theorem holds for n = k and consider the case n = k + 1. In view of the condition (2.37, k+1 (q k+1 q k+1 = k i (q i q i. (2.39 The definition of a regular sequence imlies that k+1 is not a zero divisor in the quotient R/( 1,..., k, hence it follows from eq.(2.39 that that is, there exist λ 1,...,λ k R such that i=1 q k+1 q k+1 R( 1,..., k, (2.40 q k+1 q k+1 = k λ i i. (2.41 Therefore, a comosition of transformations [i,k + 1] λ i for 1 i k turns K(;q into a Koszul matrix factorization 1, q 1 λ 1 k+1 1, q 1 λ 1 k+1 K.. = K.. R K( k+1 ;q k+1 (2.42 k, q k λ k k+1 k, q k+1 k, q k λ k k+1 Since i=1 k i (q i λ i k+1 = i=1 k i q i, (2.43 then by the inductive assumtion 1, q 1 λ 1 k+1 1, q K.. 1 = K.. (2.44 k, q k λ k k+1 k, q k and the tensor roduct in the r.h.s. of eq.(2.42 is isomorhic to K(;q, which roves the theorem. Consider three olynomials: W Q[x] and,q Q[x,y]. Let us fix a ositive integer n such that Let M be a matrix factorization of the olynomial i=1 deg y < n. (2.45 W = W (y n q. (2.46

17 16 In view of the condition (2.45, the quotient M = M/(y n M (2.47 is a free module over Q[x]. Hence it reresents a matrix factorization of the olynomial W over that algebra. A tensor roduct of M with a Koszul matrix factorization of a secial form is also a matrix factorization of W. M = K(y n ;q Q[x,y] M (2.48 Theorem 2.2. M and M are homotoy equivalent as matrix factorizations over Q[x] via a natural homotoy equivalence morhism M f M. (2.49 Proof. The module of the tensor roduct M = K(y n ;q Q[x,y] M is the sum of modules M M 1, and its twisted differential is the sum of twisted differentials of the modules and the homomorhisms in the diagram where D is the twisted differential of M. y n D M 1 M D, (2.50 The matrix factorization M contains a subfactorization as demonstrated by the following diagram q M = K ( 1; (y n q Q[x,y] M, (2.51 M M 1 M M (y n q y n q M M y n (2.52 Its to line reresents M, its bottom line reresents M and the vertical homomorhisms reresent the injection. It is obvious from the diagram (2.52 that M/ M = M/(y n M = M. The Koszul matrix factorization K ( 1; (y n q is contractible, hence M is also contractible and M = M/ M. Together with the revious isomorhism, this roves the homotoy equivalence M = M. The equivalence is established by the quotient ma M f M/ M = M, (2.53

18 17 which is natural in M. Different homotoy equivalence mas f commute with each other. Namely, consider five olynomials: W Q[x], 1 Q[x,y 1 ] and 2,q 1,q 2 Q[x,y 1,y 2 ] with the conditions Let M be a matrix factorization of the olynomial deg y1 1 < n 1, deg y2 2 < n 2. (2.54 W = W (y n q 1 (y n q 2. (2.55 Its tensor roduct with two Koszul matrix factorizations M = K(y n ;q 1 Q[x,y1,y 2 ] K(y n ;q 2 Q[x,y1,y 2 ] M (2.56 is a matrix factorization of W and we consider it over the algebra Q[x]. A double alication of Theorem 2.2 leads to the homotoy equivalence where M M, (2.57 M = M/ ( (y n M + (y n M (2.58 is a matrix factorization of W over Q[x]. However, one could establish the homotoy equivalence (2.57 by either alying Theorem 2.2 first to y 1, 1,q 1 and then to y 2, 2,q 2 or in the oosite order, thus obtaining two homotoy equivalence homomorhisms M f,2 f,1 f,1 f,2 M (2.59 Theorem 2.3. The homotoy equivalence homomorhisms (2.59 are equal: f,2 f,1 = f,1 f,2. (2.60 Proof. The matrix factorization M in (2.56 has the structure M = (y n M q 2 M 1 y n q 1 y n q 1 y n M 1 M q 2 (2.61

19 18 This matrix factorization has a subfactorization M : M = (y n M q 2 M 1 y n q 1 y n q 1 y n M 1 q 2 (y n M + (y n M (2.62 A double alication of the roof of Theorem 2.2 to M indicates that M is contractible and both comositions f,2 f,1 and f,1 f,2 are canonical homomorhisms between a module and its quotient: M f,2 f,1 f,1 f,2 M/ M = M. ( Grahs, tangles and associated matrix factorizations. In our categorification constructions, an oen grah is a lanar grah diagram with secial univalent vertices called legs., and are examles of oen grahs. The legs are enumerated and oriented as in or out, but sometimes we dro these decorations on the ictures. If an oen grah has no legs, then it is called closed. The grahs Γ r resulting from the resolution of the crossings of L are examles of closed grahs. More generally, a grah-tangle is a lanar grah diagram with legs and also with secial 4-valent vertices called crossings. These vertices are decorated with the under-over selection for the incident edges. A link diagram L and an elementary tangle are examles of grah-tangles, the first one being, in fact, closed. There are two oerations on the sets of closed grahs (and similarly on grah-tangles. The first oeration is a disjoint union (γ 1,γ 2 γ 1 γ 2 (2.64 We call two legs of an oen grah or a grah-tangle ˆγ close if they can be connected by a line in the comlement of ˆγ. The second oeration joins two close legs i and j with oosite orientations: γ # i,j # i,j (γ. (2.65

20 19 Our categorification is a ma from oen grahs to matrix factorizations and from grahtangles to comlexes of matrix factorizations γ ˆγ, τ ˆτ. (2.66 The categorification ma should convert the oerations (2.64 and (2.65 into aroriate algebraic counterarts. First of all, we choose a basic set of variables x = (x 1,...,x m and a basic olynomial W(x Q[x]. To a set l of n enumerated oriented legs (of an oen grah or a grah-tangle we associate a olynomial of nm variables according to the formula x l = x 1,...,x n, x i = x 1,i,...,x m,i, (2.67 W l (x l = n µ i W(x i, (2.68 i=1 where µ i is the orientation of the i-th leg: µ i = 1 if it is oriented outwards, and µ i = 1 if it is oriented inwards. To an oen grah γ we associate a matrix factorization ˆγ Ob (MF Wl(γ, (2.69 where l(γ is a set of oriented legs of γ. Similarly, to a grah-tangle τ we associate a comlex of matrix factorizations u to homotoy: ( ˆτ Ob K(MF Wl(τ. (2.70 If Γ is a closed grah then l(γ =, so W l(γ = 0, and as a matrix factorization, ˆΓ is isomorhic to its own homology where by definition (cf. eq.(1.17. ˆΓ C (Γ, (2.71 C (Γ = H MF (ˆΓ (2.72 Finally, we secify the functoriality rules for the mas (2.66. To a disjoint union of oen grahs γ 1 γ 2 we associate the tensor roduct of matrix factorizations γ 1 γ 2 = ˆγ 1 ˆγ 2, (2.73 and to the joining of legs we associate the identification of the variables x i and x j : # i,j (γ = ĥi,j(ˆγ, (2.74

21 20 where ĥi,j is the functor (2.17 corresonding to the algebra homomorhism Q[x l(γ ] h i,j Q[x l(γ ]/(x 1,i x 1,j,...,x m,i x m,j. (2.75 It is imortant to note that the r.h.s. of eq.(2.74 should be considered as a matrix factorization over the ring Q[x l(#i,j (γ], that is, the multilication by x i and x j is not a art of the module structure, since these variables are no longer related to legs. Nevertheless, ˆx i and ˆx j remain collections of endomorhisms of #i,j (γ, and in view of the quotient (2.75. ˆx i = ˆx j (2.76 The functorial roerties of the categorification ma allow us to define it first for the elementary oen grahs and grah-tangles, which are just single vertices together with their legs, and for the 1-arc grah, and then derive the categorification of more comlicated objects by cutting them into elementary ieces and assembling the corresonding matrix factorizations with the hel of eqs.(2.73 and (2.75. The matrix factorization is determined by the rincial roerty of : gluing it to a leg of another grah γ would roduce the same grah. In other words, if γ has n legs enumerated by numbers 1...n, then # i,n+1 (γ n+1 n+2 = γ, 1 i n, (2.77 where γ is the same grah as γ, excet the i-th leg of γ is labeled as (n + 2-th. Then the formulas (2.73 and (2.75 require that ( ˆγ xi Q[xi ] i n+2 ˆγ xn+2. (2.78 Note that the roerty (2.78 of the 1-arc matrix factorization allows us to resent the r.h.s. of eq.(2.74 as a tensor roduct with 1-arc matrix factorization, so that eq.(2.74 can be written as # i,j (γ ˆγ ( Q[xi,x j ] i j. (2.79 This formula is similar to definition of the Hochschild homology of a bimodule, laying the role of the algebra resolution. 3. Arc matrix factorizations ( i j 3.1. Basic olynomial. We begin the categorification of the SO(2N + 2 Kauffman olynomial by choosing the basic algebra Q[x,y] and the basic olynomial W(x,y = xy 2 + w(x, w(x = x 2N+1, (3.1

22 21 which was reviously considered by Gukov and Walcher [2]. This olynomial has a homogeneous q-degree if we set deg q W(x,y = 4N + 2, (3.2 deg q x = 2, deg q y = 2N, (3.3 and this allows us to use graded matrix factorizations for categorification. In articular, the twisted differential should have a homogeneous q-degree deg q D = 2N + 1. (3.4 The basic olynomial (3.1 is an odd function of its combined argument: W( x, y = W(x,y. (3.5 This allows us to imose an additional constraint on the categorification ma (2.66: if γ is obtained from γ by reversing the orientation of the i-th leg, then ˆγ is obtained from ˆγ by changing the signs of the variables x i,y i, that is, ˆγ x i,y i = ˆγ xi, y i. (3.6 Therefore, from now on we assume that all legs of grahs and tangles are oriented outwards, unless secified otherwise, and whenever we need to join two legs, we change the orientation of one of them with the hel of eq. (3.6. In other words, the homomorhism h i,j in the joining formula (2.74 should be modified from eq.(2.75 to Q[x l(γ ] h i,j Q[x l(γ ]/(x 1,i + x 1,j,...,x m,i + x m,j, (3.7 because i-th and j-th legs are both oriented outwards. We introduce a convenient notation for iterated differences of w(x, which we define recursively as For examle, w(x 1,...,x n 1,x n = w(x 1,...,x n 2,x n 1 w(x 1,...,x n 2,x n x n 1 x n. (3.8 w(x 1,x 2 = w(x 1 w(x 2 x 1 x 2, w(x 1,x 2,x 3 = w(x 1,x 2 w(x 1,x 3 x 2 x 3. (3.9 Note that all these olynomials are symmetric functions of their arguments.

23 Jacobi algebra. The Jacobi algebra of W is defined as the quotient J W = Q[x,y]/( x W, y W = Q[x,y]/(y 2 + (2N + 1x 2N,xy. (3.10 It is graded with a q-graded basis 1,x...x 2N,y, deg q x i = 2i, deg q y = 2N. (3.11 Hence the dimensions of q-graded subsaces of J W are 1 if 0 i 2N and i N, dimj W,{2i} = 2 if i = N, (3.12 and the graded dimension of J W is equal to the Kauffman olynomial of the unknot (1.4 u to a degree shift: dim q J W = q4n+2 1 q q 2N = q 2N P unkn (q. (3.13 This agreement indicates that eq.(3.1 is a suitable choice for the basic olynomial. Let 1,x...(x 2N,y J W (3.14 denote the dual basis of (3.11, and for v J W let ˆv : J v W J W denote the linear oerator of multilication by v. Then the algebra multilication ma J W J W m J W (3.15 can be resented as m = 2N i=0 ˆx i (x i + ŷ y. (3.16 The Jacobi algebra J W acquires a Frobenius algebra structure, if we choose the Frobenius trace to be 1 Tr F = 2(2N + 1 (x2n (3.17 (the origin of the normalization factor will become clear later. Then the co-multilication ma J W J W J W (3.18 can be resented as ( = 2 ŷ y (2N + 1 2N i=0 ˆx i x 2N i. (3.19

24 The 1-arc Koszul matrix factorization and the unknot sace. To a 1-arc oen grah we associate a Koszul matrix factorization ( y 2 + y 1, x 2 (y 2 y = K. (3.20 x 2 + x 1, y1 2 + w( x 1,x 2 of the olynomial W 2 = W(x 1,y 1 + W(x 2,y 2. (3.21 The left column of this Koszul matrix factorization contains sums rather than differences, because in our conventions both legs of the 1-arc grah 1 2 are oriented outwards. Theorem 2.2 indicates that the matrix factorization (3.20 satisfies the roerty (2.78. Also note that in view of eq.(2.23, ˆx 2 ˆx 1, ŷ 2 ŷ 1. (3.22 According to eq.(2.26, the dual to the 1-arc matrix factorization is the matrix factorization of the 1-arc grah in which both legs are oriented inwards: ( 1 2 = 1 2{2N}. (3.23 The unknot diagram can be constructed by joining legs 1 and 2 of the arc 1 2, hence, in accordance with eqs.(1.17 and (2.74, the corresonding sace C unkn is the homology of the differential vector sace constructed by setting in the Koszul matrix factorization (3.20: x 2 = x 1 = x, y 2 = y 1 = y (3.24 R 00 y 2 +w (x R 01 {1 2N} 2xy 2xy R 00 (3.25 (y 2 +w (x R 11 { 2N} R 10 { 1} R11 { 2N} ( 2xy This is a Koszul comlex K. Since the olynomials xy,y 2 + w (x form a y 2 + w (x regular sequence, the homology of the comlex (3.25 aears only in the lower right corner and, in fact, as a q-graded vector sace it is isomorhic to the Jacobi algebra (3.10 u to a q-degree shift: H i MF ( = W J, if i = 0, 0, if i = 1, where J W = J W { 2N}. (3.26

25 24 Thus C unkn = J W, deg Z2 C unkn = 0. (3.27 It follows from this equation and from eq.(3.13 that the relation (1.11 holds for the crossingless diagram of the unknot. C unkn is isomorhic to J W not only as a graded vector sace but also as a module over J W whose action on C unkn is generated by ˆx = ˆx 1 = ˆx 2, ŷ = ŷ 1 = ŷ 2. (3.28 This module structure allows us to introduce a convenient basis of C unkn. Let the first basis element e be a non-zero element of the 1-dimensional lowest q-degree subsace C unkn,{ 2N} C unkn, (3.29 then the rest of the basis is generated by the action of the elements of J W on e: e, ˆx(e, ˆx 2 (e... ˆx 2N (e,ŷ(e C unkn. (3.30 Later we will endow C unkn with an algebra structure, and this will turn (3.27 into an algebra isomorhism. Then e will be chosen to be the unit of J W. Finally, we describe the endomorhism algebra of the 1-arc matrix factorization (3.20. The artial derivatives form a regular sequence, hence x W = y 2 + (2N + 1x 2N, y W = 2xy (3.31 End MF ( 1 2 = Ext 0 ( 1 2, 1 2 = J W, Ext 1 ( 1 2, 1 2 = 0. (3.32 This isomorhism follows directly from (3.26. Indeed, according to the relations (2.16, (3.23 and to the joining legs formula (2.79, Ext i ( 1 2, 1 2 = H i MF( 1 2 R ( 1 2 = H i MF( 1 2 R J = H i W if i = 0, MF ( {2N} = 0 if i = 1, {2N} (3.33 where R = Q[x 1,y 1,x 2,y 2 ]. The isomorhism J W = End MF ( 1 2 can be generated by the homomorhism ˆ in two different ways: (x,y (ˆx 1,ŷ 1 or (x,y (ˆx 2,ŷ 2. (3.34 Both versions differ by a sign in view of eq.(3.22.

26 25 The calculations (3.33 leading to eq. (3.32 can be generalized to find the morhisms between multi-arc grahs. Theorem 3.1. Consider a system of 2m distinct 1-vertices (legs. Let γ and γ be two oen grahs which are the unions of m 1-arc grahs connecting these 2m legs. Then Ext 0 (γ,γ = (J W m 0 {2N(m m 0 }, Ext 1 (γ,γ = 0, (3.35 where m 0 is the number of circles formed by joining γ and γ together through their common legs. Proof. Let γ be the grah γ, in which we reversed the orientations of all legs, so that they are now oriented inwards. Then Ext i (ˆγ, ˆγ = H i MF(ˆγ R (ˆγ = H i MF(ˆγ R ˆγ {2Nm}, (3.36 and the formula (3.35 follows from the basic roerty of the 1-arc matrix factorization and from eq.( arc Koszul matrix factorizations Category, endo-functors and saddle morhisms. Four legs 1, 2, 3, 4 can be connected by two arcs in three different ways: ( The third connection requires the arcs to intersect in the lane of the diagram, however when we assign the matrix factorization to the oen grah, we treat this intersection as non-existent (that is, virtual, while considering to be a disjoint union of two arcs similar to and. The matrix factorizations,, belong to the same category of matrix factorizations of the olynomial over the algebra W 4 (x,y = 4 W(x i,y i (3.38 i=1 R = Q[x,y], x = x 1,...,x 4, y = y 1,...,y 4. (3.39

27 26 The structure of 2-arc matrix factorizations is simle: in view of eq.(2.73 they are tensor roducts of matrix factorizations corresonding to constituent arcs. In articular, the virtual nature of the crossing imlies that = (3.40 The symmetric grou S 4 acts on the set of grahs (3.37 by ermuting their legs, e.g. σ 12. (3.41 The olynomial (3.38 is invariant under the simultaneous ermutations of the variables x and y, hence the elements σ S 4 act as endo-functors on the category MF W4. Theorem 3.1 describes the saces Ext i (ˆγ, ˆγ for 2-arc grahs γ and γ : J Ext 0 (ˆγ, ˆγ W J W if γ = γ, = Ext 1 (ˆγ, ˆγ = 0. (3.42 J W {2N} if γ γ, Suose that γ γ. According to eq.(3.42, dim Ext 0 {2N} (ˆγ, ˆγ = 1. (3.43 Therefore, u to a constant factor, there exists a unique non-trivial morhism ˆγ F ˆγ, deg q F = 2N. (3.44 We call it the saddle morhism and we denote it either as F, or G, or H deending on the choice of γ and γ. The uniqueness of the saddle morhism (u to a constant factor for a given air of grahs γ γ imlies that the elements of S 4 transform saddle morhisms into saddle morhisms (these functors are images of grou elements, hence they are invertible and can not transform a saddle morhism into a trivial morhism Proer 2-arc Koszul matrix factorizations. We have to know an exlicit resentation of a saddle morhism to erform comutations with it. According to eqs.(3.20 and (3.40, the 2-arc matrix factorizations ˆγ are resented as 4-row Koszul matrix factorizations, so their modules have rank (8,8, which makes exlicit formulas for morhisms rather unwieldy. However, as we shall see, all 2-arc matrix factorizations can be resented as tensor roducts ˆγ = K cmn R ˆγ, (3.45

28 27 where K cmn is a common 2-row Koszul matrix factorization (the same for all ˆγ, while ˆγ are roer 2-row Koszul matrix factorizations. The saddle morhisms act as identity on K cmn, that is ˆγ F ˆγ (3.46 K cmn R ˆγ id R F K cmn R ˆγ for a suitable ma ˆγ F ˆγ. In order to describe this construction exlicitly, we introduce a roer algebra R = Q[,q,r,C], where = ( 1, 2, q = (q 1,q 2, r = (r 1,r 2, (3.47 and a roer olynomial We also define a homomorhism by the formulas and where W 4, = 1 q 2 r 2 + q 1 2 r 2 + r 1 2 q q 1 r 1 C. (3.48 R h R (3.49 h ( 1 = x 2 + x 4, h (q 1 = x 3 + x 4, h (r 1 = x 1 + x 4, h ( 2 = y 2 + y 4, h (q 2 = y 3 + y 4, h (r 2 = y 1 + y 4. (3.50 h (C = C(x, (3.51 C(x = w(x 1,x 2, x 3,x 4 + w(x 1, x 1,x 2,x 4 + w(x 1, x 1, x 2,x 4 (3.52 and eq.(3.8 defines the olynomial w. Let S 3 S 4 be the subgrou ermuting the legs (1, 2, 3. Consider its ermutation action on the trilet of variables (r,,q. Note that the formulas (3.50 are invariant with resect to the simultaneous action of S 3 on (x 1,x 2,x 3, (y 1,y 2,y 3 and (r,,q. The olynomial W 4, is invariant under the action of S 3 on (r,,q, hence the elements σ S 3 act as endo-functors ˆσ on the category of matrix factorizations MF R,W 4,. In articular,

29 28 they ermute the matrix factorizations ( 1, q 2 r 2 + q 1 r 1 C = K, (3.53 2, q 2 r 1 + q 1 r 2 ( r 1, 2 q q 1 C = K, (3.54 r 2, 1 q q 1 ( q 1, 2 r r 1 C = K, (3.55 q 2, 1 r r 1 equivariantly with resect to their action on the 2-arc grahs: ˆσ(ˆγ = (σ(γ. (3.56 Define ˆγ = ĥ(ˆγ, (3.57 where ĥ is the functor (2.17 induced by the homomorhism (3.49. Proosition 3.2. Three 2-arc matrix factorizations factor as where and ˆγ = K cmn R ˆγ, (3.58 ( x 1 + x 2 + x 3 + x 4, A(x,y K cmn = K y 1 + y 2 + y 3 + y 4, B(x,y (3.59 A(x,y = (y 3 + y 4 (y 1 + y 2 + y 4 y 1 y 2 + w( x 1,x 3 + (x 2 + x 4 w(x 1,x 2, x 3 (3.60 (x 2 + x 4 (x 1 + x 4 ( w(x 1, x 1,x 2,x 4 + w(x 1, x 1, x 2,x 4 B(x,y = x 1 y 1 + x 2 y 2 + x 3 y 3 x 4 y 4. (3.61 Proof. We will rove this roosition for γ = According to our definitions,, the roofs for other grahs are similar. K cmn R = x 1 + x 2 + x 3 + x 4, A(x,y y 1 + y 2 + y 3 + y 4, B(x,y K x 2 + x 4, (y 1 + y 4 (y 3 + y 4 + (x 1 + x 4 (x 3 + x 4 C(x (3.62 y 2 + y 4, (x 1 + x 4 (y 3 + y 4 + (y 1 + y 4 (x 3 + x 4

30 29 A straightforward comutation shows that this is a matrix factorization of the olynomial W 4 (x,y. On the other hand, the matrix factorization is a 4-row Koszul matrix factorization of the same olynomial, which is transformed by the combination [3, 1] 1 [4, 2] 1 of the transformations (2.34 into the following form x 1 + x 3, y1 2 + w( x 1,x 3 y 1 + y 3, x 3 (y 3 y 1 = K x 2 + x 4, y2 2 + w( x 2,x 4 y 2 + y 4, x 4 (y 4 y 2 x 1 + x 2 + x 3 + x 4, y1 2 + w( x 1,x 3 y 1 + y 2 + y 3 + y 4, x 3 (y 3 y 1 = K x 2 + x 4, y2 2 y1 2 + w( x 2,x 4 w( x 1,x 3. y 2 + y 4, x 4 (y 4 y 2 x 3 (y 3 y 1 (3.63 The latter Koszul matrix factorization shares the same first column with the matrix factorization (3.62. Since the olynomials in that column form a regular sequence, the claim of the roosition follows from Theorem Saddle morhisms 4.1. A roer saddle morhism. According to the formulas (2.31 and (2.32, the roer matrix factorization ˆγ has an exlicit form R 2 1 P i R 2 0 Q i R 2 1, (4.1 where the homomorhisms P i and Q i are resented by matrices ( ( 2 1 q 1 r 2 + q 2 r 1 1 P =, Q =, (4.2 q 2 r 2 + q 1 r 1 C (q 2 r 1 + q 1 r 2 q 2 r 2 + q 1 r 1 C 2 ( ( r 2 r 1 1 q q 1 r 1 P =, Q 2 q q 1 C ( 1 q q 1 =, (4.3 2 q q 1 C r 2 ( ( q 2 q 1 1 r r 1 q 1 P =, Q =. (4.4 2 r r 1 C ( 1 r r 1 2 r r 1 C q 2

31 30 Consider the following homomorhism between two matrix factorization modules R 2 1 P R 2 0 Q R 2 1, (4.5 where F R 2 1 F 1 P R 2 0 F 0 Q ( ( 0 1 q 2 q 1 F 0 =, F q2 2 q1c 2 1 =. (4.6 0 q 1 C q 2 By using exressions (4.2 and (4.3, it is easy to verify that the diagram (4.5 is commutative, that is, [D,F ] = 0. This means that F defines a matrix factorization morhism. The isomorhisms (3.45 allow us to extend F to a morhism between full 2-arc matrix factorizations: F, where F = id R F, F = ĥ(f (4.7 R 2 1 F 1 (cf. diagram (3.46. Proosition 4.1. The morhism (4.7 is a saddle morhism. Proof. It is easy to verify that deg Z2 F = deg Z2 F = 0, deg q F = deg q F = 2N, (4.8 so it remains to verify that F 0. Indeed, the matrix F 0 has a unit entry, so the matrix of F also has a unit entry. However, all entries of the twisted differential matrices (4.2 and (4.3 belong to the ideal generated by the variables x 1,...,x 4,y 1,...,y 4, hence there does not exist a homomorhism X Hom R (,, such that F = [D,X]. The saddle morhisms for other airs of 2-arc grahs γ γ can be obtained from eqs.(4.6 and (4.7 by the simultaneous ermutation action of S 3 on the legs of the grahs and on the variables (r,, q Comositions of saddle morhisms. Let us find a comosition of two saddle morhisms ˆγ F ˆγ G ˆγ, γ γ,γ. (4.9 We should consider two different cases: γ = γ and γ γ. In both cases the symmetric grou S 4 acts transitively on such trilets of 2-arc grahs (γ,γ,γ, so it is sufficient to

32 31 erform the calculation just for one trilet, and the answer for others could be deduced by ermutation of legs and variables. Proosition 4.2. The comosition of two saddle morhisms F H (4.10 is equal to HF 2ŷ 1 ŷ 2 2(2N + 1 2N i=0 ˆx i 1 ˆx 2N i 2. (4.11 Proof. Consider the comosition of roer saddle morhisms R 2 1 P R 2 0 Q R 2 1 (4.12 F R 2 1 F 1 P R 2 0 F 0 Q R 2 1 F 1 H R 2 1 H 1 P R 2 0 H 0 Q R 2 1 H 1 Since H = ˆσ 12 (F, and σ 12 switches r and while leaving q and C intact, we conclude from eq.(4.6 that and matrix multilication shows that Hence H 0 = F 0, H 1 = F 1, (4.13 H F = (q 2 2 q 2 1C id. (4.14 HF = (ŷ 3 + ŷ 4 2 (ˆx 3 + ˆx 4 2 C(ˆx, (4.15 where C is defined by eq.(3.52. Now it is an elementary algebra exercise to transform the r.h.s. of this formula into the r.h.s. of eq.(4.11 with the hel of eqs.(3.52, (3.1 and the following relations between the endomorhisms of : ˆx 3 ˆx 1, ŷ 3 ŷ 1, ˆx 4 ˆx 3, ŷ 4 ŷ 3, (4.16 ŷ 1 (2N + 1 ˆx 2N 1, ŷ 2 (2N + 1 ˆx 2N 2. (4.17

33 32 Proosition 4.3. The comosition of two saddle morhisms F G (4.18 is null-homotoic: GF 0. (4.19 Proof. By definition, eq.(4.19 means that there exists an R-module homomorhism X, (4.20 such that deg Z2 X = 1, deg q X = 2N 1 (4.21 and GF = {D,X} (4.22 (we ut a minus sign in for future convenience. Consider the roer saddle morhisms R 2 1 P R 2 0 Q R 2 1 (4.23 F R 2 1 F 1 P R 2 0 F 0 Q R 2 1 F 1 G R 2 1 G 1 P R 2 0 G 0 Q R 2 1 G 1 where the second morhism is resented by the matrices ( ( G 0 =, G =, (4.24 1C 0 1 C 2

34 33 in accordance with the relation G = ˆσ 13ˆσ 12 (F. On the diagram G 1 F 1 R 2 1 P R 2 0 X 0 G 0 F 0 P Q R 2 1 X 1 G 1 F 1 Q (4.25 R 2 1 R 2 0 R 2 1 reresenting the comosition of saddle morhisms we choose an R-module homomorhism X Hom R (,, X = X 0 + X 1 (4.26 resented by the matrices ( ( q X 0 =, X 1 =. (4.27 q 1 C 0 1 C 2 A direct comutation shows that these matrices satisfy the relations G 0 F 0 = (P X 0 + X 1 Q, G 1 F 1 = (Q X 1 + X 0 P, (4.28 which means that X satisfies the roerties and if we choose deg Z2 X = 1, deg q X = 2N 1, G F = {D,X }, (4.29 X = id R X, (4.30 then X would satisfy (4.21 and ( Semi-closed and closed saddle morhisms. In order to exhibit the structure of the categorification comlex C (L and to rove its Reidemeister move invariance we have to find what haens to the saddle morhism when we join together one or two airs of legs of the 2-arc oen grahs. As a result of this joining, a 2-arc grah becomes either an 1-arc grah, or a 1-arc grah with a disjoint circle, or a circle, or a air of circles. Hence the initial 2-arc matrix factorizations can be simlified by homotoy equivalence transformations, and we want to know how this simlification affects the saddle morhisms.

35 Semi-closed saddle morhisms. Let us join one air of legs in the saddle morhisms. The endomorhism symmetry S 4 allows us to consider only three cases, namely, the saddle morhisms F H G, (4.31 in which we join together legs 2 and 4. After that the diagram becomes F H G, (4.32 its homomorhisms resulting from taking the quotient of the saddle homomorhisms (4.31 by the ideal (x 2 + x 4,y 2 + y 4 according to eq.(3.7. The second and the third grahs in the diagram (4.32 are just 1-arcs, so we ass to homotoy equivalent matrix factorizations and resent this diagram as Cunkn F H G. (4.33 The first matrix factorization in this diagram slits into a direct sum of 1-arc matrix factorizations 2N ( ( Cunkn = ˆx (e i y, (4.34 i=0 where ˆx i (e and y form the basis (3.30 of the sace C unkn. Hence we will exress the semi-closed saddle morhisms F and G in terms of the endomorhisms of the 1-arc matrix factorization 3. According to eq.(3.33, where J W is generated by 1 ( 3 End MF = J W, ( ˆx = ˆx 1 = ˆx 3, ŷ = ŷ 1 = ŷ 3. (4.36 Recall that the sace C unkn has a secial basis (3.30. Let us introduce the dual basis for the dual sace C unkn : e, (ˆx(e, (ˆx 2 (e... (ˆx 2N (e, (ŷ(e C unkn. (4.37

36 35 Proosition 4.4. With the aroriate choice of the basis element e, the homomorhisms F and G of the diagram (4.33 are homotoic to the following homomorhisms ( 2N F F m = ˆx i (ˆx i (e (ŷ(e + ŷ, (4.38 H H = 2 i=0 The homomorhism G is null-homotoic: ( ŷ ŷ(e (2N + 1 2N i=0 ˆx i ˆx 2N i (e. (4.39 G 0. (4.40 The indices m and indicate that the homomorhisms F m and H are related to the multilication (3.16 and comultilication (3.19 in J W, as we will see shortly. Proof. We derive the exressions for the semi-closed saddle morhisms not by a direct comutation, but rather from three roerties of the saddle morhisms (4.31. First, a saddle morhism has homogeneous q-degree 2N, so and these morhisms must have a form F H 2N i=0 2N i=0 deg q F = deg q H = 2N. (4.41 a i ˆx i (ˆx i (e + ã ŷ (ŷ(e + a 1 ŷ (ˆx N (e + a 2ˆx N ( ŷ(e, (4.42 b i ˆx i (ˆx 2N i (e + b ŷ (ŷ(e + b 1 ŷ (ˆx N (e + b 2 ˆx N ( ŷ(e, (4.43 where the coefficients a s and b s are rational numbers. The second roerty is the comosition formula (4.11, which we aly to the comosition H F : H F 2ŷ 1 ŷ 2 2(2N + 1 ˆx2N+1 1 ˆx 2N+1 2. (4.44 ˆx 1 ˆx 2 It imoses relations among the coefficients of exressions (4.42, which imly that F af m, H bh, a,b Q, ab = 2. (4.45 Indeed, if we aly eq.(4.44 to e, then we find that H bh and a 0 b = 2. The other coefficients of (4.42 are determined by alying the relation (4.44 to ˆx i (e and to ŷ(e. Since ab = 2, then a 0 and we can rescale the basis element e so that a = 1, b = 2. Thus we obtain eqs.(4.38 and (4.39.

37 36 Finally, Proosition 4.3 says that G F 0. (4.46 At the same time, according to eq.(4.38, the morhism F acts on the submodule 3 e 3 C unkn as id, so G must be null-homotoic Closed saddle morhisms. Let us consider the result of joining legs 2 and 4, as well as legs 1 and 3 in the grahs of the diagram (4.31: F H G. (4.47 The first grah in this diagram is a air of disjoint circles, while the second and the third grahs are circles. Therefore, we can use the matrix factorization homotoy equivalence in order to resent the diagram (4.47 as C unkn C unkn F H C unkn G C unkn. (4.48 The morhisms on this diagram can be obtained from the morhisms of the diagram (4.33 by imosing the joining relation ˆx 1 = ˆx 3 in the exressions (4.38, (4.39 and (4.40. In fact, since ˆx 3 never aears there, the exressions remain the same. First of all, we find that G = 0. (4.49 Second, we observe that the oerators ˆx and ŷ aearing in the formulas for F m and H are the same as the oerators ˆx and ŷ of eqs.(3.16 and (3.19. Thus, if we establish a canonical isomorhism by the basis elements corresondence then the diagram (4.48 becomes C unkn = J W = J W { 2N} (4.50 ˆx i (e x i, ŷ(e y, (4.51 J W J W m J W 0 J W, (4.52

38 37 while the homomorhisms (4.38 and (4.39 take the form F F m = H H = 2 2N i=0 ( ˆx i (x i + ŷ y, (4.53 ŷ y (2N + 1 2N i=0 ˆx i x 2N i. ( Multi-dimensional Postnikov systems and their convolutions The category of matrix factorizations is triangulated. Relation (4.19 means that the diagram (4.18 is a chain comlex of matrix factorization morhisms. This allows us to define the matrix factorization as its convolution. But first let us review the basic facts about Postnikov systems and their convolutions. The general theory of Postnikov systems within triangulated categories is described in the book [1] (Chater IV, exercises. We adat its aroach to the secific case of matrix factorizations. Some roofs in this section are omitted, because they are standard exercises in homological algebra A multi-dimensional Postnikov system. Let us introduce multi-index notations: for a ositive integer n and v i Z we denote v = (v 1,...,v n and v = n i=1 v i. Also v w means that v i w i for all i = 1...n, and v > w means that v w and v w. The zero vector is denoted 0 = (0,...,0, and the vectors e 1,...,e n form the standard basis in the lattice Z n. We introduce a new Z n grading, which we call length, with multi-degree deg lng = (deg 1,...,deg n, (5.1 and assume that deg lng x = 0 for all x R. We also introduce the total length degree: deg lng = deg lng. (5.2 A length-graded R-module A is called bounded if its grading exansion A = k Z n A k, deg lng A k = k (5.3 has only finitely many non-trivial modules A k. A homomorhism X Hom R (A,B between two length-graded R-modules is called non-negative if it is a sum of homomorhisms of non-negative length degrees: X = k 0 X k, deg lng X k = k. (5.4 Let Hom R; 0 (A,B denote the sace of all such homomorhisms.