1. On generalization of Khovanov link homologies
Krzysztof Putyra
Jagiellonian University, Kraków
Geometry & Topology of Manifolds
Kraków, 4th July 2008

2. Knots & diagrams f Knot: a smooth embedding f : S1 R3.
Knot space: X = {f : S1R3 | f is a knot}
Funkcja z okręgu w R^3  „analisys situs”
Założenia: gładka (C1), albo PL
Równoważność: leżenie w jednej składowej łukowe
Diagramy: rzut na płaszczyznę, spełnia pewne warunki
Twierdzenie i ruchy Reidemeistera
Two knots are equivalent iff they lie in the same path component of the knot space X.
In similar way we define links as smooth embeddings of disjoint union of circles..

3. Knots & diagrams p tunnels bridge
This gives us a diagram of a knot/link.

4. Knots & diagrams
unknot
trefoil
figure-eight knot
cinquefoil

5. Knots & diagrams

6. Knots & diagrams R1 R2 R3 Kurt Reidemeister
Theorem (K. Reidemeister, 1927).
Let L1, L2 be links with diagrams D1, D2. Then L1, L2 are equivalent iff D1 can be obtained from D2 by a finite sequence of moves, called Reidemeister moves:
Kurt Reidemeister
R1 R2 R3
and isotopies of the plane.

7. Khovanov Complex
0-smoothing
1-smoothing
Mikhail Khovanov

8. direct sums create the complex
Khovanov Complex 3 1
110
101
100
010
001 2
arrows are cobordisms
objects are smoothed diagrams
000 –
111
Mikhail Khovanov
011
direct sums create the complex
C-3
C-2
C-1 C0 d

9. direct sums create the complex
Odd Khovanov Complex 3 1
110
100 2
arrows are cobordisms
objects are smoothed diagrams with arrows
000
111
101
010
Mikhail Khovanov
011
001
direct sums create the complex
(applying some edge
assignment)
C-3
C-2
C-1 C0 d

10. ORS ‘half-proj.’ functor
Khovanov functor
see Khovanov: arXiv:math/
FKh: Cob → ℤ-Mod
symmetric:
Edge assignment is given explicite.
ORS ‘half-proj.’ functor
see Ozsvath, Rasmussen, Szabo: arXiv:
FORS: ArCob → ℤ-Mod
not symmetric:
Edge assignment is given by homological properties.
Category of cobordisms is symmetric:

11. Main question
Fact (Bar-Natan) Invariance of the Khovanov complex can be proved at the level of topology.
Question Can Cob be changed to make FORS
a functor?
Motivation Invariance of the odd Khovanov complex may be proved at the level of topology and new theories may arise.
Dror Bar-Natan
Anwser Yes: cobordisms with chronology

12. ChCob: cobordisms with chronology & arrows
Chronology τ is a Morse function with exactly
one critical point over each critical value.
Critical points of index 1 have arrows:
τ defines a flow φ on M
critical point of τ are fix points φ
arrows choose one of the in/outcoming
trajectory for a critical point.
Chronology isotopy is a smooth homotopy H satisfying:
- H0 = τ0
- H1 = τ1
- Ht is a chronology

13. ChCob: cobordisms with chronology & arrows
Chronology is preserved:
Critical points do not vanish:

14. F: ChCob ℤ-Mod Which conditions should a functor
satisfy to produce homologies?

15. Chronology change condition
This square needs to be anti-commutative after multiplying some egdes with invertible elements (edge assignment proccess).
These two compositions could differ by an invertible element only!

16. Chronology change condition
The coefficient should be well-defined for any change of chronology: α α α α3
= α2β α α β

17. Chronology change condition
The necessary conditions are the following:
where X 2 = Y 2 = 1 and Z is invertible.
Note (X, Y, Z) → (-X,-Y,-Z) induces an isomorphism on complexes.

18. Edge assignment P = λrP = λrλf P = ... = Π λiP P = λrP P
Proposition For any cube of resolutions C(D) there exists an edge assignment e → φ(e)e making the cube anticommutative.
Sketch of proof Each square S corresponds to a change of chronology with some coefficient λ. The cochain
ψ(S) = -λ
is a cocycle: 6
P = λrP = λrλf P = ... = Π λiP
P = λrP P
P = λrP = λrλf P
i = 1
By the ch. ch. condition:
dψ(C) = Π -λi = 1
and by the contractibility of a 3-cube:
ψ = dφ 6
i = 1

19. Edge assignment
Proposition For any cube of resolutions C(D) there exists an edge assignment e → φ(e)e making the cube anticommutative.
Proposition For any cube of resolutions C(D) different egde assign-ments produce isomorphic complexes.
Sketch of proof Let φ1 and φ2 be edge assignments for a cube C(D). Then
d(φ1φ2-1) = dφ1dφ2-1 = ψψ-1 = 1
Thus φ1φ2-1 is a cocycle, hence a coboundary. Putting
φ1 = dηφ2
we obtain an isomorphism of complexes ηid: C(D,φ1) →
C(D,φ2).

20. Edge assignment
Proposition For any cube of resolutions C(D) there exists an edge assignment e → φ(e)e making the cube anticommutative.
Proposition For any cube of resolutions C(D) different egde assign-ments produce isomorphic complexes.
Corollary Denote by D1 and D2 a link diagram D with different choices of arrows. Then the cubes of resolutions C(D1) and C(D2) are isomorphic.
Corollary Upto isomophisms the Khovanov complex C(D) depends only on the link diagram D.

21. compare with Bar-Natan: arXiv:math/0410495
S / T / 4Tu relations
compare with Bar-Natan: arXiv:math/
Theorem The Khovanov complex is invariant under chain homotopies and the following relations:
where X, Y and Z are given by the ch.ch.c.

22. Main result Theorem There exists a functor
satisfying ch.ch.c and S/T/4Tu, where
FU: ChCob R-Mod
R = ℤ[X, Y, Z±1]/(X2 = Y2 = 1)
Moreover:
If (X,Y,Z) = (1,1,1), then FU is the Khovanov functor
If (X,Y,Z) = (1,-1,1), then FU is the ORS functor

23. Main result M FU (M) Rv+⊕Rv– R = ℤ[X, Y, Z±1]/(X2 = Y2 = 1) 1  v+
v+  ZY v– 
v+  v–  1 v+
  X v+  v+
  Z-1 v– v–  v+ v–
  Z v+ v+  v– v–
  Y v–  v+
  v+ v+
  v– v– v–
  ZX v– v+ v–
  0 v–  v+ v+  v– v– 

24. (X, Y, Z) → (-X, -Y, -Z) and (X, Y, Z) → (Y, X, Z-1)
Further research
There exist isomorphisms of complexes
(X, Y, Z) → (-X, -Y, -Z) and (X, Y, Z) → (Y, X, Z-1)
Hence we have three theories over ℤ: Khovanov, ORS and (-1, -1, 1).
Is the last different or not?
The proof of independence goes for any functor F: ChCob → R-Mod.
Does there exist a transformation between such functors F and G?
If so, when is it an equivalence?
How powerful is the odd Khovanov complex? Does it detect the unknot?
What about tangles?

25. Thank you for your attention