Understanding Khovanov Homologies: Exploring Foley's Theorem

Even more oddKhovanov Homologies Krzysztof Putyra Jagiellonian University, Krakow CATG, Gdańsk 2008 13th June 2008

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

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

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

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. Anwser Yes: cobordisms with chronology

ChCob: cobordisms with chronology & arrows Chronology is a Morse function with exactly one critical point over each critical value. Critical points of index 1 have arrows. Chronology isotopy is a smooth homotopy H satisfying: - H0 = τ0 - H1= τ1 - Ht is a chronology

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

Which conditions should a functor F: ChCob ℤ-Mod 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 = α2β α3 α α β

Chronology change condition The necessary conditions are as follows: where X 2 = Y 2 = Z 2 = 1 Note (X,Y,Z) → (-X,-Y,-Z) induces isomorphism on complexes.

By ch. ch. condition: ψ(S) = Π-λi = 1 and by contractibility: ψ = dφ 6 i = 1 Edge assignment Proposition There is an edge assignment e → φ(e)e making the cube anticommutative. Sketch of proofEach square S corresponds tochange 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

S / T / 4Tu relations compare with Bar-Natan: arXiv:math/0410495 Theorem The complex is invariant of a link under homotopy and following 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.

Main result Theorem There 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: • (X,Y,Z) = (1,1,1), then FU is the Khovanov functor (with c = 0) • (X,Y,Z) = (1,-1,1), then FU is the ORS functor

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–

Thank you Handout’s URL: http://www.math.toronto.edu/~drorbn/People/Putyra/GWU08-handout.pdf

Understanding Khovanov Homologies: Exploring Foley's Theorem