1. A 2-category of dotted cobordisms and a universal odd link homology
Krzysztof Putyra
Columbia University, New York
III Knots in Poland, Będlewo
July 27, 2010

2. What is covered?
Even vs odd link homologies Chronological cobordisms Dotted cobordisms with chronologies Chronological Frobenius algebras

3. Cube of resolutions A crossing has two resolutions
Type 0 (up)
Type 1 (down)
Example A 010-resolution of the left-handed trefoil
Louis Kauffman 1 2 3 1 2 3
Dwa wygładzenia skrzyżowania:
skrzyżowanie
dwa wygładzenia
intuicja geometryczna
010

4. vertices are smoothed diagrams
Cube of resolutions
110
101
011
100
010
001 3 1 2
vertices are smoothed diagrams
000
111
edges are cobordisms
Kostka wygładzeń dla trójlistnika
000
drogi elementarne (jedna zmiana)
kostka
Observation This is a commutative diagram in a category of 1-manifolds and cobordisms

5. Khovanov complex Even homology (K, 1999) Apply a graded functor
Odd homology (O R S, 2007)
see: arXiv:math/
see: arXiv:
Apply a graded functor
Apply a graded pseudo-functor
Mikhail Khovanov
Peter Ozsvath
funktor / pseudo-funktor
Result: a cube of modules with commutative faces
Result: a cube of modules with both commutative and anticommutative faces

6. direct sums create the complex
Khovanov complex
Odd: signs given by homological properties
Even: signs given explicitely
direct sums create the complex
{+1+3}
{+2+3}
{+3+3}
{+0+3}
Mikhail Khovanov
Peter Ozsvath
jak uzyskać antyprzemienność
Theorem Homology groups of the complex C(D) are link invariants. The graded Euler characte-ristic of C(D) is the Jones polynomial JL(q).

7. edges are cobordisms with signs
Khovanov complex
000
100
010
001
110
101
011
111 1 2 3
edges are cobordisms with signs
Dror Bar-Natan
Objects: sequences of smoothed diagrams
Morphisms: „matrices” of cobordisms
Theorem (B-N, 2005) The complex is a link invariant under chain homotopies and some local relations.

8. Khovanov complex Odd homology (P, 2008) = + – = {-1}  {+1} = 1 = 0
Even homology (B-N, 2005)
Odd homology (P, 2008)
Complexes for tangles in Cob Dotted cobordisms: Neck-cutting relation: Delooping and Gauss elimination: Lee theory:
Complexes for tangles in ChCob ? ?? ??? ????
= –
Upraszczamy kobordyzmy, by zmniejszyć kompleks
even: dotted cobordisms (Dror, 2006)
neck-cutting relation
delooping
dotted are universal
odd - ?
= {-1}  {+1}
= 1
= 0

9. Chronological cobordisms
A chronology: a separative Morse function τ.
An arrow: choice of a in/outcoming trajectory of a gradient flow of τ
Pick one
Chronologia jako funkcja Morsa, odróżnienie różnych ścieżek w kostce
An isotopy of chronologies: a smooth homotopy H s.th. Ht is a chronology
Fact If τ0  τ1 and dimW = 2, there exist isotopies of M and I that induce an isotopy of these chronologies.

10. Chronological cobordisms
A change of a chronology is a smooth homotopy H. Changes H and H’ are equivalent if H0  H’0 and H1  H’1.
Remark Ht might not be a chronology for some t (so called critical moments).
Fact (Cerf, 1970) Every homotopy is equivalent to a homotopy with finitely many critical moments of two types:
type I:
type II:
Zmiany chronologii pozwalają zmieniać ścieżki
Zmiany a rozszerzenie kategorii (uniwersalne rozszerzenie przy założeniu ścisłej symetrii)
Theorem (P, 2008) 2ChCob with changes of chronologies is a 2-cate-gory. This category is weakly monoidal with a strict symmetry.

11. Chronological cobordisms
Critical points cannot be permuted: 
Critical points do not vanish:
Zmiany chronologii pozwalają zmieniać ścieżki
Zmiany a rozszerzenie kategorii (uniwersalne rozszerzenie przy założeniu ścisłej symetrii)
Arrows cannot be reversed:

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

13. Chronological cobordisms
A solution in an R-additive extension for changes:
type II: identity
generic type I:
MM = MB = BM = BB = X X2 = 1
SS = SD = DS = DD = Y Y2 = 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) = (, ).

14. Chronological cobordisms
Some of the changes:
where X 2 = Y 2 = 1
Note (X, Y, Z) → (-X, -Y, -Z) induces an isomorphism on complexes.

15. Chronological cobordisms
A solution in an R-additive extension for changes:
type II: identity
generic type I:
exceptional type I:
MM = MB = BM = BB = X X2 = 1
SS = SD = DS = DD = Y Y2 = 1
SM = MD = BS = DB = Z
MS = DM = SB = BD = Z-1
AB = X Y Z   bdeg(A) = (, ) bdeg(B) = (, )
even odd
XYZ
YXZ
ZYX
1 / XY
X / Y

16. Chronological cobordisms
A solution in an R-additive extension for changes:
type II: identity
general type I:
exceptional type I: 1 / XY or X / Y
Theorem (P, 2010) With the above:
Aut(W) = {1} if #hdls(W) = 0 and #sphr(W)  1
Aut(W) = {1, XY} otherwise
MM = MB = BM = BB = X X2 = 1
SS = SD = DS = DD = Y Y2 = 1
SM = MD = BS = DB = Z
MS = DM = SB = BD = Z-1
AB = X Y Z   bdeg(A) = (, ) bdeg(B) = (, )
even odd
XYZ
YXZ
ZYX

17. Chronological cobordisms
compare with Bar-Natan: arXiv:math/
Theorem (P, 2008) The complex is invariant under Reidemeister moves up to chain homotopies and the following local relations:
where the critical points on the shown parts of cobordisms are consequtive, i.e. any other critical point appears earlier or later than the shown part.

18. Dotted chronological cobordisms
Motivation Cutting a neck due to 4Tu:
I may be 0!
Z(X+Y) =
Add dots formally and assume the usual S/D/N relations:
= 0
(S)
(N)
= –
= 1
(D)
bdeg(  ) = (-1, -1)
I’m homo-geneous!
What is a dot?
Odd neck-cutting relation
Changes of chronologies and dots
A chronology takes care of dots, coefficients may be derived from (N):
M = B = XZ
S = D = YZ-1
 = XY M =

19. Dotted chronological cobordisms
Motivation Cutting a neck due to 4Tu:
I may be 0!
Z(X+Y) =
Add dots formally and assume the usual S/D/N relations:
= 0
(S)
I’m homo-geneous!
= 1
(D)
bdeg(  ) = (-1, -1)
(N)
= –
What is a dot?
Odd neck-cutting relation
Changes of chronologies and dots
A chronology takes care of dots, coefficients may be derived from (N):
M = B = XZ S = D = YZ-1  = XY
Remark T and 4Tu can be derived from S/D/N.
Notice all coefficients are hidden!

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

21. Dotted chronological cobordisms
Theorem There are isomorphisms Mor(, )  R[h, t]/((XY – 1)h, (XY – 1)t) =: R Mor(, )  v+R  v-R =: A given by
bdeg(h) = (-1, -1)
bdeg(t) = (-2, -2)
bdeg(v+) = ( 1, 0)
bdeg(v- ) = ( 0, -1)
h 
v+
v- 
t  
The dotted algebra
almost Lie-type theory
A is a bimodule over R :
left module:
right module:
 =
 =

22. Dotted chronological cobordisms
Algebra/coalgebra structure: given by cobordisms
= XZ =
= XZ = 
= Z2 = 
- bimoduł na A
Operations are right-linear, but not left-linear!

23. Universality of dotted cobordisms
A chronological Frobenius system (R, A) in A is given by a monoidal 2-functor F: 2ChCob  A:
R = F()
A = F( )
We further assume:
R is graded, A = Rv+ Rv is bigraded
bdeg(v+) = (1, 0) and bdeg(v) = (0, -1)
A base change: (R, A)  (R', A') where A' := A R R'
A twisting: (R, A)  (R, A')
 ' (w) =  (yw)
' (w) = (y-1w)
where y  A is invertible and deg(y) = (1, 0).
The dotted algebra
almost Lie-type theory
Theorem If (R, A') is a twisting of (R, A) then
C(D; A')  C(D; A)
for any diagram D.

24. Universality of dotted cobordisms
Theorem (P, 2010) Any rank 2 chronological Frobenius system with generators in degrees (1, 0) and (0, -1) arises from (R, A) by a base change and a twisting. Here, R = [X, Y, Z1]/(X2-1,Y2-1).
Corollary Having a chronological Frobenius system F = (RF, AF), the homology HF(L) is a quotient of H(L).
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
The dotted algebra
almost Lie-type theory

25. Even vs Odd Odd homology (P, 2010) = + – = {-1}  {+1} = 1 = 0
Even homology (B-N, 2005)
Odd homology (P, 2010)
Complexes for tangles in Cob Dotted cobordisms: Neck-cutting relation: Delooping and Gauss elimination: Lee theory:
Complexes for tangles in ChCob Dotted chronological cobordisms - only one dot over a field, if X  Y Neck-cutting with no coefficients Delooping – yes Gauss elimination – sign problem Lee theory exists only for X = Y
= –
Upraszczamy kobordyzmy, by zmniejszyć kompleks
even: dotted cobordisms (Dror, 2006)
neck-cutting relation
delooping
dotted are universal
odd - ?
= {-1}  {+1}
= 1
= 0

26. Further remarks
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 holds for the dotted version
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 with a strict symmetry in the sense of A.Beliakova and E.Wagner
A linear solution for chronological nested cobordisms exists and is given by 9 parameters (squares of 3 of them are equal 1)
idea: what should be in general a chronological multi-graded algebra
virtual: double-or-triple graded?