On generalization of odd link homologies

1. On generalizationof odd link homologies Krzysztof Putyra Jagiellonian University, Kraków Columbia University New York, 26th September 2008

2. 0-smoothing 1-smoothing Mikhail Khovanov Cube of resolutions

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

4. 110 100 000 111 101 010 001 d d d C-3 C-2 C-1 C0 Mikhail Khovanov Cube of resolutions 1 3 2 arrows are cobordisms objects are smoothed diagrams with arrows 011 direct sums create the complex (applying some edge assignment)

5. Khovanov functor see Khovanov: arXiv:math/9908171 FKh: Cob→ ℤ-Mod symmetric: Edge assignment is given explicite. Category of cobordisms is symmetric: ORS ‘projective’ functor see Ozsvath, Rasmussen, Szabo: arXiv:0710.4300 FORS: ArCob→ ℤ-Mod notsymmetric: Edge assignment is given by homological properties.

6. Main question Fact (Bar-Natan) Invariance of the Khovanov complex can be proved at the level of topology. QuestionCan 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

7. ChCob: cobordisms with chronology & arrows Chronologyτis 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

8. Chronology is preserved: Critical pointsdo not vanish: ChCob: cobordisms with chronology & arrows

9. with the full set of relations given by: ChCob: cobordisms with chronology & arrows TheoremThe category 2ChCob is generated by the following:

10. ChCob(B)‘s form a planar algebra with planar operators: ChCob(B): cobordisms with corners For tangles we need cobordisms with corners: • both input and output has same • endpoints set B • projection is a chronology • choose orientation for each • critical point • all up to isotopies preserving π • being a chronology

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

12. 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!

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

14. 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.

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

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

17. Edge assignment PropositionFor any cube of resolutions C(D) there exists an edge assignment e → φ(e)e making the cube anticommutative. PropositionFor any cube of resolutions C(D) different egde assign-ments produce isomorphic complexes. PropositionDenote by D1 and D2 a tangle diagram D with different choices of arrows. Then there exist edge assignments φ1 and φ2 s.th. complexes C(D1, φ1) and C(D2 , φ2) are isomorphic. Corollary Upto isomophisms the complex Kh(D) depends only on the tangle diagram D.

18. S / T / 4Tu relations compare with Bar-Natan: arXiv:math/0410495 TheoremThe complex Kh(T) is invariant under chain homotopies and the following relations: where X, Y and Z are given by the ch.ch.c.

19. Main result TheoremThere exists a functor satisfying ch.ch.c and S/T/4Tu, where FU: ChCobR-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

20. Main result v+  X v+ v+  v+ v+ Z-1 v– v–  v+ v– Z v+ v+  v– v–  Y v– v–  v– v+  v+ v+ v+ v– v– v–  0 v– v– ZX v– v+ v–  v+ v+  v– v–  v–

21. Further research • There exists an isomorphism of complexes • (X, Y, Z) → (-X, -Y, -Z) • Hence we have 4 theories over ℤ: Khovanov, ORS, (-1, 1, 1) and (-1, -1, 1). • Alexander Shumakovitch conjectures that ORS and (-1,1,1) are • isomorphic. 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 complex Kh(T)? Does it detect the unknot? • What about tangle cobordisms?

22. Thank youfor your attention