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Graph Homologies and Functoriality

Ahmad Zainy Al-Yasry Higher Structures in Lisbon 27.07.2017

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 1 / 31

Goal

Open a door between Graph(knot) homology and applications in Dynamical Systems and Noncommutative Geometry. Tools: Additive category (Obj Embedded graphs in the 3-sphere, Mor geometric correspondences given by 3-manifold M branched coverings of the 3-sphere along embedded graphs (or in particular knots) in the 3-sphere). Kauffman’s invariant of Graphs. Khovanov Homology for graphs and Floer Homology for Graphs. PL Cobordisms between graphs and Smooth Cobordisms between family of links.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 2 / 31

1

We construct an additive category where objects are embedded graphs in the 3-sphere and morphisms are geometric correspondences given by 3-manifolds realized in different ways as branched covers of the 3-sphere, up to branched cover cobordisms, by considering a 3-manifold M realized in two different ways as a covering of the 3-sphere as a correspondence between the branch loci (Graphs) of the two covering maps. G ⊂ S3

πG

← − M

πG′

− → S3 ⊃ G′

2

We consider dynamical systems obtained from associated convolution algebras endowed with time evolutions defined in terms of the underlying geometries.

3

We describe the relevance of our construction to the problem of spectral correspondences in noncommutative geometry.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 3 / 31

Example

(Poincaré homology sphere): Let P denote the Poncaré homology sphere. This smooth compact oriented 3-manifold is a 5-fold cover of S3 branched along the trefoil knot (that is, the (2,3) torus knot), or a 3-fold cover of S3 branched along the (2,5) torus knot, or also a 2-fold cover of S3 branched along the (3,5) torus knot.

K denote the category whose objects Obj(K ) are graphs G ⊂ S3 and whose

morphisms Mor(K ) a 3-manifold Mi with submersions πG and πG′ to S3

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 4 / 31

The composition M ◦ ˜ M

Definition

Suppose given G ⊂ S3

πG

← − M

πG′

− → S3 ⊃ G′

and G′ ⊂ S3 ˜

πG′

← − ˜

M

˜ πG′′

− → S3 ⊃ G′′.

One defines the composition M ◦ ˜ M as fibered product M ×G′ ˜ M M ◦ ˜ M := M ×G′ ˜ M = {(x,x′) ∈ M × ˜ M|πG′(x) = ˜

πG′(x′)}., ˆ

M = M ×G′ ˜ M

P1

ttttttttttt

P2

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

M

πG

①①①①①①①①

πG′

• ❑

❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ˜

M

˜ πG′

ssssssssss

˜ πG′′

• G ⊂ S3

G′ ⊂ S3 G′′ ⊂ S3

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 5 / 31

Cobordism of branched cover by Hilden and Little 1980

Two compact oriented 3-manifolds M1 and M2 that are branched covers of S3, with covering maps π1 : M1 → S3 and π2 : M2 → S3, respectively branched along 1-dimensional simplicial complex E1 and E2. A cobordism of branched coverings is a 4-dimensional manifold W with boundary

∂W = M1 ∪−M2 (where the minus sign denotes the change of orientation),

endowed with a submersion q : W → S3 × [0,1], with M1 = q−1(S3 × {0}) and M2 = q−1(S3 × {1}) and q|M1 = π1 and q|M2 = π2. The map q is a covering map branched along a surface S ⊂ S3 × [0,1] such that

∂S = E1 ∪−E2, with E1 = S ∩(S3 × {0}) and E2 = S ∩(S3 × {1}).

Two morphisms M1 and M2 in Hom(G,G′), of the form G ⊂ E1 ⊂ S3 πG,1

← − M1

πG′,1

− → S3 ⊃ E′

1 ⊃ G′

G ⊂ E2 ⊂ S3 πG,2

← − M2

πG′,2

− → S3 ⊃′

2⊃ G′.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 6 / 31

S ⊂ S3 × [0,1]

q

← − W

q′

− → S3 × [0,1] ⊃ S′,

branched along surfaces S,S′ ⊂ S3 × [0,1]. The maps q and q′ have the properties that M1 = q−1(S3 × {0}) = q′−1(S3 × {0}) and M2 = q−1(S3 × {1}) = q′−1(S3 × {1}), with q|M1 = πG,1, q′|M1 = πG′,1, q|M2 = πG,2 and q′|M2 = πG′,2. The surfaces S and S′ have boundary ∂S = E1 ∪−E2 and

∂S′ = E′

1 ∪−E′ 2, with E1 = S ∩(S3 × {0}), E2 = S ∩(S3 × {1}),

E′

1 = S′ ∩(S3 × {0}), and E′ 2 = S′ ∩(S3 × {1}). Here By “surface" we mean a

2-dimensional simplicial complex that is PL-embedded in S3 × [0,1], with boundary

∂S ⊂ S3 × {0,1} given by 1-dimensional simplicial complexes, i.e. embedded graphs.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 7 / 31

Convolution algebra and time evolution

Lemma

The set of compact oriented 3-manifolds G forms a regular semigroupoid, whose set

• f units is identified with the set of embedded graphs.

Consider the semigroupoid ring (algebra) C[G ] of complex valued functions with finite support on G , with the associative convolution product,

(f1 ∗ f2)(M) =

∑

M1,M2∈G :M1◦M2=M

f1(M1)f2(M2). Define an involution on the semigroupoid G by Hom(G,G′) ∋ α = (M,G,G′) → α∨ = (M,G′,G) ∈ Hom(G′,G), where, if α corresponds to the 3-manifold M with branched covering maps then α∨ corresponds to the same 3-manifold taken in the opposite order.

Lemma

The algebra C[G ] is an involutive algebra with the involution f ∨(M) = f(M∨).

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 8 / 31

Time evolution

Given an algebra A over C, a time evolution is a 1-parameter family of automorphisms

σ : R → Aut(A ). There is a natural time evolution on the algebra C[G ] obtained as

follows.

Lemma

Suppose given a function f ∈ C[G ]. Consider the action defined by

σt(f)(M) := n

m

it

f(M), where M a covering with the covering maps πG and πG′ respectively of generic multiplicity n and m. This defines a time evolution on C[G ]. Given a representation ρ : A → End(H ) of an algebra A with a time evolution σ, one says that the time evolution, in the representation ρ, is generated by a Hamiltonian H if for all t ∈ R one has

ρ(σt(f)) = eitHρ(f)e−itH,

for an operator H ∈ End(H ).

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 9 / 31

Convolution algebras and 2-semigroupoids

Lemma

The data of embedded graphs in the 3-sphere, 3-dimensional geometric correspondences, and 4-dimensional branched cover cobordisms form a 2-category

G 2.

We denote the compositions of 2-morphisms by the notation horizontal (fibered product): W1 ◦ W2 vertical (gluing): W1 • W2.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 10 / 31

Vertical and horizontal time evolutions

We obtain a convolution algebra associated to the 2-semigroupoid This space of functions can be made into an algebra A (G 2) with the associative convolution product

• f the form

(f1 • f2)(W) =

∑

W=W1•W2

f1(W1)f2(W2), which corresponds to the vertical composition of 2-morphisms, namely the one given by the gluing of cobordisms. Similarly, one also has on A (G 2) an associative product which corresponds to the horizontal composition of 2-morphisms, given by the fibered product of cobordisms, of the form

(f1 ◦ f2)(W) =

∑

W=W1◦W2

f1(W1)f2(W2). We also have an involution compatible with both the horizontal and vertical product structure.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 11 / 31

Vertical and horizontal time evolutions

We say that σt is a vertical time evolution on A (G 2) if it is a 1-parameter group of automorphisms of A (G 2) with respect to the product structure given by the vertical composition of 2-morphisms namely

σt(f1 • f2) = σt(f1)• σt(f2).

Similarly, a horizontal time evolution on A (G 2) satisfies

σt(f1 ◦ f2) = σt(f1)◦ σt(f2).

Vertical time evolution: Hartle-Hawking gravity. Vertical time evolution: gauge moduli and index theory. Horizontal time evolution: bivariant Chern character.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 12 / 31

Kauffman’s invariant of Graphs 1989

1

Let G be a graph embedded in a 3-manifold M.

2

Associate to G a family of knots and links prescribes that we should make a local replacement as in figure to each vertex in G.

3

A vertex v connects two edges and isolates all other edges at that vertex, leaving them as free ends.

Figure: local replacement to a vertex in the graph G

Define T(G) to be the family of the links associated to the graph G.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 13 / 31

Theorem

Let G be any graph embedded in S3, and presented diagrammatically. Then the family

• f knots and links T(G), taken up to ambient isotopy, is a topological invariant of G.

For example, in the figure the graph G2 is not ambient isotopic to the graph G1, since T(G2) contains a non-trivial link.

, { }

2 1

(G T )={ , } G G 2 T(G )=

1

, ,

Figure: family of knots and links

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 14 / 31

Khovanov-Kauffman Homology for Graphs

(Khovanov 2000) D

Khovanov

− → [D] = C∗,∗(D)

Homology

− →

KH∗,∗(D) Let G be an embedded graph, T(G) = {L1,L2,....,Ln} the family of links associated to G Then the Khovanov-Kauffman homology for the embedded graph G is given by KKh(G) = Kh(L1)⊕ Kh(L2)⊕ ....⊕ Kh(Ln)

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 15 / 31

Example

In figure T(G1) = {,} KKh(G1) = Kh()⊕ Kh() KKh(G1) = Kh()⊗ Kh()⊕ Kh() T(G2) = {

,} then

KKh(G2) = Kh(

)⊕ Kh()

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 16 / 31

Floer-Kauffman Homology for Knots

Ozsváth - Szabó and Rasmussen around 2003 Introduced Floer Homology which is an invariant of knots and links in three manifolds. Let K ⊂ S3 be an oriented knot, there are several different variants of the knot Floer homology of K. The simplest is the hat version, which takes the form of a bi-graded, finitely generated Abelian group

• HFK(K) =
• i,s∈Z
• HFK i(K,s)

Here, i is called the Maslov (or homological) grading, and s is called the Alexander

• grading. The graded Euler characteristic of

HFK is the Alexander-Conway polynomial

∑

s,i∈Z

(−1)iqsrankZ(

HFK i(K,s)) = △K(t)

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 17 / 31

Example

We used kauffman technique to introduce Khovanov-Kauffman homology for embedded graphs.

, { }

2 1

(G T )={ , } G G 2 T(G )=

1

, ,

Figure: Family of links associated to a graph

T(G1) = {,}

• HFG(G1) =

HFK()⊕ HFK() Now,

• HFG(G1) =

HFK()⊗ HFK()⊕ HFK()

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 18 / 31

Example

T(G2) = {

,} then

• HFG(G2) =

HFK(

)⊕

HFK()

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 19 / 31

Graph Cobordance

Definition

Two links L and L′ are called cobordic if there is a surface Σ have the boundary

∂Σ = L ∪−L′ with L = Σ∩(S3 × {0}), L′ = Σ∩(S3 × {1}). Here by "surfaces" we

mean 2-dimensional compact differentiable manifold embedded in S3 × [0,1]. We define the identity cobordism to be idL for a link L. [ΣL] denotes the cobordism class of the link L.

Definition

Two graphs G and G′ are called cobordic if there is a PL surface ΣP have the boundary ∂ΣP = G ∪−G′ with G = Σ∩(S3 × {0}), G′ = Σ∩(S3 × {1}). Here by "surfaces" we mean 2-dimensional simplicial complexes that are PL-embedded in S3 × [0,1]. We define the identity cobordism to be idG for a graph G. [ΣP] denotes the cobordism class of the graph G.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 20 / 31

Definition

Let G,G′ two embedded graphs have associated families of links T(G) and T(G′) respectively according to Kauffman construction, and let T(Σα)α∈λ be a family of smooth cobordisms have links boundaries of T(G) and T(G′). These graphs are said to be cobordant if there is a family of smooth cobordisms T(Σα)α∈λ in S3 × [0,1] such that T(G) and T(G′) are the boundary of T(Σα).

Theorem

The family of smooth cobordisms T(Σα)α∈λ is associated to the PL cobordism ΣP.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 21 / 31

Composition of Cobordisms

We can define the composition between two PL cobordisms ΣP and Σ′

P (where

ΣP is the cobordism with boundary G ∪−G′, Σ′

P is the cobordism with boundary

G′ ∪−G′′ in S3 × [0,1]) to be ΣP ◦ Σ′

P which is a PL cobordism with boundary

G ∪−G′′. For the second type of the cobordance by family of smooth cobordisms we can define the composition as follows : Let G,G′ and G′′ be embedded graphs in S3 with T(G),T(G′) and T(G′′) links families associated to each graph respectively. We have a family of smooth cobordisms T(Σα)α∈λ have boundary T(G)∪−T(G′) and another family T(Σ′

β)β∈γ with boundary T(G′)∪−T(G′′).

Let Σβ and α ∈ λ be a smooth cobordism with boundary links from the sets T(G) and T(G′) and let Σ′

β for β ∈ γ be a smooth cobordism with boundary links from

the sets T(G′) and T(G′′). Σα ◦ Σ′

β is a smooth cobordism with boundary links

from the sets T(G) and T(G′′) and this defines the composition of the second type of the graphs cobordance.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 22 / 31

Let GK be a category, whose objects are embedded graphs that have a family of links according to the kauffman definition and morphisms are 2-dimensional simplicial complex surface ΣP ∈ Hom(G,G′) with graphs boundary.

Lemma GK is a pre-additive category.

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 23 / 31

Functoriality

We want to study the functoriality between the Category of Graphs with the 3-manifolds branched cover GK as morphisms and the category of Floer-Kauffman Homology for Graphs with the linear maps a morphisms C FKh. We need to study the existence of a Functor F between GK and C FKh = V where V is the category of Vector Spaces, such that: For G ∈ GK

F

− →

HFG(G) ∈ C FKh Morphisms: Branched cover Spaces M

F

− → L linear maps

Composition: For M and ˜ M to morphisms in GK we want to show

F(M ◦ ˜

M) = F(M)◦ F( ˜ M)

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 24 / 31

Let G ⊂ S3

π

← − M π−1 − → S3 ⊃ G′.

We need to find a map between the two Floer Homology groups (M,π,π−1)

• HFK(G,S3)

Υ(M,π,π−1)

− →

• HFK(G′,S3)

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 25 / 31

(Sketch) We need to use the generators and boundaries explicitly

• CFK ∗(G,S3)

∂∗

• Φ1

CFK ∗(π−1(G),M)

∂∗

• CFK ∗−1(G,S3)

Φ1

CFK ∗−1(π−1(G),M)

• CFK ∗(G)

Ψ2

− →

CFK ∗(G ∪ G′)

Ψ3

− →

CFK ∗(G′)

• CFK ∗(π−1(G′),M)

∂∗

• Φ′

4

CFK ∗(G′,S3)

∂∗

• CFK ∗−1(π−1(G′),M)

Φ′

4

CFK ∗−1(G′,S3)

• HFK(G,S3)

Υ(M,π,π−1)

− →

• HFK(G′,S3)

where

Υ(M,π,π−1) = Φ′

4 ◦ Ψ3 ◦ Ψ2 ◦ Φ1

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 26 / 31

ˆ

M = M ×G′ ˜ M

P1

ttttttttttt

P2

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

M

πG

①①①①①①①①

πG′

• ❑

❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ˜

M

˜ πG′

ssssssssss

˜ πG′′

• G ⊂ S3

G′ ⊂ S3 G′′ ⊂ S3

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 27 / 31

• CFK ∗(G,S3)

∂∗

• Φ1

CFK ∗(P1 ◦ π−1(G),M × ˜ M)

∂∗

• CFK ∗−1(G,S3)

Φ1

CFK ∗−1(P1 ◦ π−1(G),M × ˜ M)

• CFK ∗(G)

Ψ2

− →

CFK ∗(G ∪ G′)

Ψ3

− →

CFK ∗(G′)

• CFK ∗(P2 ◦ ˜

πG′′,M × ˜

M)

∂∗

• Φ′

4

CFK ∗(G′′,S3)

∂∗

• CFK ∗−1(P2 ◦ ˜

πG′′,M × ˜

M)

Φ′

4

CFK ∗−1(G′′,S3)

• HFK(G,S3)

Υ(Mט

M,P1◦π,P2◦˜

π)

− →

• HFK(G′,S3)

where

Υ(M× ˜

M,P1◦π,P2◦˜

π) = Φ′

4 ◦ Ψ3 ◦ Ψ2 ◦ Φ1

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 28 / 31

Consider M in Hom(G,G′) specified by a diagram G ⊂ E ⊂ S3

π1

← − M

π2

− → S3 ⊃ E′ ⊃ G′.

We can choose W = M × [0,1] as a cobordism of M with itself. This has

∂W = M ∪−M with covering maps

G ⊂ S3 × [0,1]

q|M×{0}

← − W = M × [0,1]

q|M×{1}

− → S3 × [0,1] ⊃ G′

branched along the PL surfaces ΣP in S3 × [0,1] with ∂ΣP = G ∪−G′. The branched covering maps q|M×{0} = π1 and q|M×{1} = π2 have the properties that M = q−1

1 (S3 × {0}) = q−1 1 (S3 × {1})

π−1

1 (G) ⊂ M × {0}

π1

• M×[0,1]

ΣM

M3 × {1} ⊃ π−1

2 (G′)

π2

• G ⊂ S3 × {0}

S3×[0,1] ΣP

S3 × {1} ⊃ G′

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 29 / 31

FKh(π−1

1 (G))

φ

• fFKh

ΣM

FKh(π−1

2 (G′))

φ

• KKh(G)

fKKh ΣP

KKh(G′)

If we use the idea of Kauffman of associating family of links to each graph and use the family of smooth cobordisms to the PL cobordism, and hence we can think by a another functor call it ψ from the link Floer Homology category to the link Khovanov Homology category, which takes the object HFK((L)) for a link L to the object Kh(L) and morphism fFh to the morphism fKh T(π−1

1 (G))

π1

• T(Sα)α∈λ

T(π−1

2 (G′))

π2

• T(G)

T(Σα)α∈λ T(G′)

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 30 / 31

Thank you!

Ahmad Zainy Al-Yasry () Graph Homologies and Functoriality Higher Structures in Lisbon 27.07.2017 31 / 31