1. XXX Knots in Washington 21 st 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 Why 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. A crossing has two resolutions Example A 010 -resolution of the left-handed trefoil Louis Kauffman Type 0 (up)Type 1 (down) 1 2 31 2 3 010

4. A change of a resolution is a cobordism Put a saddle over the area being changed:

5. 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

6. 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

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

8. Idea:  Stay in Cob as long as possible!  Build a complex in  - Cob  Prove it is invariant Applications:  Natural extension over tangles  A categorification of the Jones polynomial for tangles  Planar algebra of complexes  Faster computations for nice links Dror Bar-Natan

9. 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

10. A chronology: a separative Morse function τ. An isotopy of chronologies: a smooth homotopy H s.th. H t is a chronology Conjecture Every isotopy of chronologies is induced by an isotopy of the cobordism and an isotopy of an interval. An arrow: choice of a in/outcoming trajectory of a gradient flow of τ Pick one Almost Theorem Every isotopy of chronologies is equivalent to one induced by an isotopy of the cobordism and an isotopy of an interval.

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

12. 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:

13. Remark Not every cobordism has a trivial automorphism group: Remark The problem does not exist in case of embedded or nested cobordisms of genus zero.

14. 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

15. 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) = ( ,  ).

16. 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

17. 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

18. 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

19. 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 = + –

20. 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) = +

21. 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!

22. 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?

23. Theorem There are isomorphisms Mor( ,  )   [X, Y, Z  1, h, t]/ ( (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 

24. 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( )

25. 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  =

26. 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

27. 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:

28. 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 )

29. 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 = XZ – +XZ h  XZ v+v+ v- v-  t  XZ  Recall: so that  (v -, v - ) = v + t + v - h

30. 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 = 

31. 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 )

32. 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))

33. A twisting: (R, A)  (R', A')  ' (w) =  (yw)  ' (w) =  (y -1 w) where y  A is invertible and 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.

34. 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 = + –

35.  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  „Categorifying categorification” – Radmila’s categorification of  [x] may be used to categorify Frobenius systems as well as this presentation