Odd homology of tangles and cobordisms
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Transcript of Odd homology of tangles and cobordisms
Odd homologyof tangles and cobordisms
Krzysztof PutyraJagiellonian University, Kraków
XXVII Knots in Washington10th January 2009
Cube of resolutions
0-smoothing 1-smoothing
Mikhail Khovano
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Cube of resolutions1
2
3
C-3 C-2 C-1 C0d d d
edges are cobordis
ms
direct sums create the complex
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vertices are smoothed diagrams –
–
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Mikhail Khovano
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Cube of resolutions1
2
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C-3 C-2 C-1 C0d d d
edges are cobordism
s with arrows
direct sums create the complex(applying
some edgeassignment
)
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vertices are smoothed diagrams
Peter Ozsvath
Khovanov functorsee 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
not symmetric:
Edge assignment is given by homological properties.
Motivation Invariance of the odd Khovanov complex may be proved at the level of topology and new theories may arise.
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?
Anwser Yes: cobordisms with chronology
Main question
Dror Bar-Natan
ChCob: cobordisms with chronology & arrowsChronology 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
Critical points cannot be permuted:
Critical points do not vanish:
ChCob: cobordisms with chronology & arrowsTheorem The category 2ChCob is generated by the following:
with the full set of relations given by:
ChCob: cobordisms with chronology & arrows
Theorem 2ChCob with changes of chronologies is a 2-category.
Change of chronology is a smooth homotopy H s.th.- H0 = τ0, H1 = τ1
- Ht is a chronology except t1,…,tn, where one of the following occurs:
ChCob(B): cobordisms with cornersFor tangles we need cobordisms with corners:
• input and output has same endpoints
• projection is a chronology• choose orientation for each
critical point• all up to isotopies preserving π
being a chronology
ChCob(B)‘s form a planar algebra with planar operators:
1 3 2 (M1,M2,M3)M1
M3
M2
Which conditions should a functor
F: ChCob ℤ-Mod
satisfies 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 conditionExtend cobordisms to formal sums over a commutative ring R.Find a representation of changes of chronology in U(R) s.th.
α M1 … Ms = β M1 … Ms => α = β
Fact WLOG creation and removing critical points can be represented by 1.Hint Consider the functor given by:
α β Id on others gen’s
Chronology change conditionExtend cobordisms to formal sums over a commutative ring R.Find a representation of changes of chronology in U(R) s.th.
α M1 … Ms = β M1 … Ms => α = β
Fact WLOG creation and removing critical points can be represented by 1.
Proposition The representation is given by
where X2 = Y2 = 1 and Z is a unit.
X Y Z
XY 1
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!
Edge assignmentProposition 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) = -λis a cocycle:
P i = 1
6
By the ch. ch. condition:
dψ(C) = Π -λi = 1
and by the contractibility of a 3-cube:
ψ = dφ
6
i = 1
P = λrPP = λrP = λrλf PP = λrP = λrλf P = ... = Π λiP
Edge assignmentProposition 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: Kh(D,φ1) →Kh(D,φ2).
Edge assignment
Proposition Denote 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.
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.
S / T / 4Tu relationscompare with Bar-Natan: arXiv:math/0410495
Theorem The complex Kh(D) is invariant under chain homotopies and the following relations:
where X, Y and Z are given by the ch.ch.c.Dror Bar-Natan
HomologiesM FXYZ (M)
Rv+⊕Rv–
1 v+
v+ + ZY v–
v+ 0 v– 1
v– Y v–
v+
v+
v+
v+
v+ Z-1v–
v–
v+
v– Z v+
v+
v–
v–
v–
v+ v+v+
v+ v–v–
v– ZX v–v+
v– 0v–
v–
v+
v+
v–
v–
v–
X
HomologiesObservation The most general ring is ℤ[X, Y, Z±1]/(X2 = Y2 = 1).
I Equivalence: (X, Y, Z) (-X, -Y, -Z)
-
and Id on others generators.
HomologiesObservation The most general ring is ℤ[X, Y, Z±1]/(X2 = Y2 = 1).
I Equivalence: (X, Y, Z) (-X, -Y, -Z)
II Equivalence: (X, Y, Z) (X, Y, 1)(X, Y, Z) (V, m, Δ, η, ε, P)(X, Y, 1) (V, m’, Δ’, η’, ε’, P’)Take φ: V V as follows: φ(v+) = v+
φ(v-) = Zv-
Define Φn: Vn Vn:Φn = φn-1 … φ id.
ThenΦ: (V, m, Δ, η, ε, P) (V, m’, ZΔ’, η’, ε’, P’)
Use now the functor given by
Z and Id on others generators
HomologiesObservation The most general ring is ℤ[X, Y, Z±1]/(X2 = Y2 = 1).
I Equivalence: (X, Y, Z) (-X, -Y, -Z)
II Equivalence: (X, Y, Z) (X, Y, 1)
Corollary There exist only two theories over an integral domain. Observation Homologies KhXYZ are dual to KhYXZ :
KhXYZ (T*) = KhYXZ(T)*
Corollary Odd link homologies are self-dual.
Tangle cobordismsTheorem For any cobordism M between tangles T1 ans T2 there exists a map
Kh(M): Kh(T1) Kh(T2)defined upto a unit.Sketch of proof (local part like in Bar-Natan’s)Need to define chain maps for the following elementary cobordisms and its inverses:
first row: chain maps from the prove of invariance theoremsecond row: the cobordisms themselves.
Tangle cobordisms
Satisfied due to the invariance theorem.
I type of moves: Reidemeister moves with inverses („do nothing”)
Tangle cobordismsII type of moves: circular moves („do nothing”)
- flat tangle is Kh-simple (any automorphism of Kh(T) is a multi-
plication by a unit)- appending a crossing preserves Kh-simplicity
Tangle cobordismsIII type of moves: non-reversible moves
Need to construct maps explicite.
Problem No planar algebra in the category of complexes: having planar operator D and chain maps f: A A’, g: B B’, the induced map
D(f, g): D(A, B) D(A’, B’)may not be a chain map!
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111011
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*001*0
11*
*11
00*01* 10*
0*1
0*0
1*1*01
F0
Local to global: partial complexes
000 100
110
111011
001
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*001*0
11*
*11
00*01* 10*
0*1
0*0
1*1*01
F0
F1
Local to global: partial complexes
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111011
001
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*001*0
11*
*11
00*01* 10*
0*1
0*0
1*1*01
F0
F*
F1
Local to global: partial complexes
Summing a cube of complexes000 100
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*001*0
11*
*11
00*01* 10*
0*1
0*0
1*1*01
F0
F*
F1
KomnF – cube of partial complexes
example: Kom2F(0) = KomF0
Proposition Komn Komm = Komm+n
Local to global: partial complexes
Tangle cobordismsBack to proof Take two tangles
T = D(T1, T2) T’ = D(T1’, T2)and an elementary cobordisms M: T1 T1’. For each smoothed diagram ST2 of T2 we have a morphism
D(Kh(M), Id): D(Kh(T1), ST2) D(Kh(T1), ST2)
- show it always has an edge assignment- any map given by one of the relation movies induced a
chain map equal Id (D is a functor of one variable)
These give a cube map of partial complexesf: KomnC(T) KomnC(T ’)
where n is the number of crossings of T2.
References1. D. Bar-Natan, Khovanov's homology for tangles and
cobordisms, Geometry and Topology 9 (2005), 1443-1499
2. J S Carter, M Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs 55, AMS, Providence, RI(1998)
3. V. F. R. Jones, Planar Algebras I, arXiv:math/9909027v1
4. M. Khovanov, A categorication of the Jones polynomial, Duke Mathematical Journal 101 (2000), 359-426
5. P. Osvath, J. Rasmussen, Z. Szabo, Odd Khovanov homology, arXiv:0710.4300v1
Thank youfor your attention