1. AMS Meeting, Newark Special Session on Homology Theories for Knots and Skein Modules 22 nd May 2010 Krzysztof Putyra Columbia University, New York

2. What are link homologies?  Cube of resolutions  Even & odd link homologies via modules via chronological cobordisms What are dotted cobordisms?  chronology on dotted cobordisms  neck-cutting relation and delooping What is a chronological Frobenius algebra?  dotted cobordisms as a baby-model  universality of dotted cobordisms with NC

3. 1 2 3 110 101 011 100 010 001 000 111 vertices are smoothed diagrams Observation This is a commutative diagram in a category of 1-mani-folds and cobordisms edges are cobordisms

4. Apply a graded functor i.e. Apply a graded pseudo-functor i.e. Peter Ozsvath Mikhail Khovanov Result: a cube of modules with commutative faces Result: a cube of modules with both commutative and anticommutative faces

5. direct sums create the complex Theorem Homology groups of the complex C are link invariants. Peter Ozsvath Mikhail Khovanov Even: signs given explicitely {+1+3} {+2+3}{+3+3} {+0+3} Odd: signs given by homological properties

6. 000 100 010 001 110 101 011 111 1 2 3 Dror Bar-Natan Theorem (2005) The complex is a link invariant under chain homotopies and relations S/T/4Tu. edges are cobordisms with signs Objects: sequences of smoothed diagrams Morphisms: „matrices” of cobordisms

7. Complexes for tangles in Cob Dotted cobordisms: Neck-cutting relation: Delooping and Gauss elimination: Lee theory: Complexes for tangles in ChCob ? ?? ??? ???? =  { -1 }   { +1 } = 1 = 0 = + –

8. A chronology: a separative Morse function τ. An isotopy of chronologies: a smooth homotopy H s.th. H t is a chronology An arrow: choice of a in/outcoming trajectory of a gradient flow of τ Pick one

9. Critical points cannot be permuted: Critical points do not vanish: Arrows cannot be reversed:

10. Theorem 2ChCob with changes of chronologies is a 2-category. This category is weakly monoidal with a strict symmetry. A change of a chronology is a smooth homotopy H. Changes H and H’ are equivalent if H 0  H’ 0 and H 1  H’ 1. Remark H t might not be a chronology for some t (so called critical moments). Fact Every homotopy is equivalent to a homotopy with finitely many critical moments of two types: type I: type II:

11. A solution in an R-additive extension for changes:  type II: identity Any coefficients can be replaced by 1 ’s due to scaling:  a a  b b

12. A solution in an R-additive extension for changes:  type II: identity  general type I: MM = MB = BM = BB = XX 2 = 1 SS = SD = DS = DD = YY 2 = 1 SM = MD = BS = DB = Z MS = DM = SB = BD = Z -1 Corollary Let bdeg(W) = (B-M, D-S). Then AB = X  Y  Z  -  where bdeg(A) = ( ,  ) and bdeg(B) = ( ,  ).

13. A solution in an R-additive extension for changes:  type II: identity  general type I:  exceptional type I: MM = MB = BM = BB = XX 2 = 1 SS = SD = DS = DD = YY 2 = 1 SM = MD = BS = DB = Z MS = DM = SB = BD = Z -1 AB = X  Y  Z  -  bdeg(A) = ( ,  ) bdeg(B) = ( ,  ) 1 / XY X / Y evenodd XYZ 1 -1 YXZ 1 -1 ZYX 1 -1

14. edges are chronological cobordisms with coefficients in R Fact The complex is independent of a choice of arrows and a sign assignment used to make it commutative. 1 2 3 000 100 010 001 110 101 011 111

15. Theorem The complex C(D) is invariant under chain homotopies and the following relations: where X, Y and Z are coefficients of chronology change relations. Dror Bar-Natan

16. Motivation Cutting a neck due to 4Tu: Add dots formally and assume the usual S/D/N relations: A chronology takes care of dots, coefficients may be derived from (N): MM = = 0 (S)(S) (N)(N) = + – = 1 (D)(D) bdeg(  ) = (- 1, - 1 )  M = B  = XZ  S = D  = YZ -1  = XY Z(X+Y) = +

17. Motivation Cutting a neck due to 4Tu: Add dots formally and assume the usual S/D/N relations: A chronology takes care of dots, coefficients may be derived from (N): Z(X+Y) = + = 0 (S)(S) (N)(N) = + – = 1 (D)(D) bdeg(  ) = (- 1, - 1 )  M = B  = XZ  S = D  = YZ -1  = XY Remark T and 4Tu can be derived from S/D/N. Notice all coefficients are hidden!

18. Theorem (delooping) The following morphisms are mutually inverse:  {–1}  {+1} – Conjecture We can use it for Gauss elimination and a divide-conquer algorithm. Problem How to keep track on signs during Gauss elimination?

19. Theorem There are isomorphisms Mor( ,  )   [X, Y, Z  1, h, t]/ ( X 2, Y 2, (XY – 1 )h, (XY – 1 )t ) =: R  Mor( , )  v + R   v - R  =: A  given by Corollary There is no odd Lee theory: t = 1  X = Y Corollary There is only one dot in odd theory over a field: X  Y  XY  1  h = t = 0 bdeg(h) = (- 1, - 1 ) bdeg(t) = (- 2, - 2 ) bdeg(v + ) = ( 1, 0 ) bdeg(v - ) = ( 0, - 1 ) h  XZ v+v+ v- v-  t  XZ 

20. Baby model: dotted algebra R  = Mor( ,  ) A  = Mor( , ) Here, F(X) = Mor( , X). A chronological Frobenius system (R, A) in A is given by a monoidal 2 -functor F: 2 ChCob  A: R = F(  ) A = F( ) Universality 

21. A chronological Frobenius system (R, A) = (F(  ), F( )) Baby model: dotted algebra (R , A  ): F(X) = Mor( , X)  weak tensor product in ChCob product in R bimodule structure on A  =

22. A chronological Frobenius system (R, A) = (F(  ), F( )) Baby model: dotted algebra (R , A  ): F(X) = Mor( , X)  weak tensor product in ChCob product in R bimodule structure on A  left productright product

23. A chronological Frobenius system (R, A) = (F(  ), F( )) Baby model: dotted algebra (R , A  ): F(X) = Mor( , X)  weak tensor product in ChCob product in R bimodule structure on A  = left module: right module:

24. A chronological Frobenius system (R, A) = (F(  ), F( )) Baby model: dotted algebra (R , A  ): F(X) = Mor( , X)  weak tensor product in ChCob  changes of chronology torsion in R symmetry of A = XY = XZ -1 = YZ -1 no dots: XZ / YZ one dot: 1 / 1 two dots: XZ -1 / YZ -1 three dots: Z -2 / Z -2 ( 1 – XY)a = 0,bdeg(a) < 0 bdeg(a) = 2 n > 0 AB = X  Y  Z  -  bdeg(A) = ( ,  ) bdeg(B) = ( ,  ) cob: bdeg: ( 1, 1 )( 0, 0 )(- 1, - 1 ) (- 2, - 2 ) ( 1, 0 ) ( 0, - 1 )

25. A chronological Frobenius system (R, A) = (F(  ), F( )) Baby model: dotted algebra (R , A  ): F(X) = Mor( , X)  weak tensor product in ChCob  changes of chronology  algebra/coalgebra structure = XZ = = = Z 2 = 

26. A chronological Frobenius system (R, A) = (F(  ), F( )) Baby model: dotted algebra (R , A  ): F(X) = Mor( , X)  weak tensor product in ChCob (right) product in R bimodule structure on A  changes of chronology torsion in R: 0 = ( 1 –XY)t = ( 1 –XY)s 0 2 = … symmetry of A:tv + = Z 2 v + thv - = XZv - h…  algebra/coalgebra structure right-linear, but not left We further assume: R is graded, A = R 1  Rα is bigraded bdeg( 1 ) = ( 1, 0 ) and bdeg(α) = ( 0, - 1 )

27. A base change: (R, A)  (R', A') where A' := A  R R' Theorem If (R', A') is obtained from (R, A) by a base change then C(D; A')  C(D; A)  R' for any diagram D. Theorem (P, 2010) Any rank two chronological Frobenius system (R, A) is a base change of (R U, A U ), defined as follows: bdeg(c) = bdeg(e) = ( 1, 1 ) bdeg(h) = (- 1, - 1 ) bdeg( 1 A ) = ( 1, 0 ) bdeg(a) = bdeg(f) = ( 0, 0 )bdeg(t) = (- 2, - 2 ) bdeg(  ) = ( 0, - 1 ) with  ( 1 ) = –c  ( 1 ) = (et–fh) 1  1 + f (YZ 1  +  1 ) + e   (  ) = a  (  ) = ft 1  1 + et( 1  + YZ -1  1 ) + (f + YZ -1 eh)  A U = R[  ]/(  2 –  h –t) R U =  [X, Y, Z  1, h, t, a, c, e, f]/(ae–cf, 1 –af+YZ -1 (cet–aeh))

28. A twisting: (R, A)  (R', A')  ' (w) =  (yw)  ' (w) =  (y -1 w) where y  A is invertible and deg(y) = ( 1, 0 ). Theorem If (R', A') is a twisting of (R, A) then C(D; A')  C(D; A) for any diagram D. Theorem The dotted algebra (R , A  ) is a twisting of (R U, A U ). Proof Twist (R U, A U ) with y = f +  e, where v + = 1 and v – = . Corollary (P, 2010) The dotted algebra (R , A  ) gives a universal odd link homology.

29. Complexes for tangles in Cob Dotted cobordisms: Neck-cutting relation: Delooping and Gauss elimination: Lee theory: Complexes for tangles in ChCob Dotted chronological cobordisms - universal - only one dot over field, if X  Y Neck-cutting with no coefficients Delooping – yes Gauss elimination – sign problem Lee theory exists only for X = Y =  { -1 }   { +1 } = 1 = 0 = + –

30.  Higher rank chronological Frobenius algebras may be given as multi-graded systems with the number of degrees equal to the rank  For virtual links there still should be only two degrees, and a punctured Mobius band must have a bidegree (–½, –½)  Embedded chronological cobordisms form a (strictly) braided monoidal 2-category; same for the dotted version unless (N) is imposed  The 2-category nChCob of chronological cobordisms of dimension n can be defined in the same way. Each of them is a universal extension of nCob in the sense of A.Beliakova