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

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..

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

Knots & diagrams unknot trefoil figure-eight knot cinquefoil

Knots & diagrams

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.

Khovanov Complex 0-smoothing 1-smoothing Mikhail Khovanov

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

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

ORS ‘half-proj.’ functor Khovanov functor see Khovanov: arXiv:math/9908171 FKh: Cob → ℤ-Mod symmetric: Edge assignment is given explicite. ORS ‘half-proj.’ functor see Ozsvath, Rasmussen, Szabo: arXiv:0710.4300 FORS: ArCob → ℤ-Mod not symmetric: Edge assignment is given by homological properties. Category of cobordisms is symmetric:

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

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

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

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

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!

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

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.

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

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).

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.

compare with Bar-Natan: arXiv:math/0410495 S / T / 4Tu relations compare with Bar-Natan: arXiv:math/0410495 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.

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

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

(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?

Thank you for your attention