Sur l'homologie sl3 des enchevêtrements ; algèbres de ... · Introduction Algebras K Super...

Introduction Algebras Kε Superficial webs Characterisation Perspectives

Sur l’homologie sl3 des enchevetrements ; algebresde Khovanov – Kuperberg

Louis-Hadrien Robert

These dirigee par Christian Blanchet

Louis-Hadrien Robert Algebre de Khovanov – Kuperberg

1 Introduction

2 The Khovanov – Kuperberg algebras

3 Superficial webs

4 A Characterisation of indecomposable web-modules

5 Perspectives

Definition

A knot is an embedding of S1 in S3.

Definition

A link is an embedding of⊔

S1 in S3.

Theorem (Freyd – Hoste – Lickorish – Millett – Ocneanu – Yetter,Przytycki – Traczyk)

There exists a unique Laurent polynomial invariant 〈·〉 satisfyingthe following relations:

⟨ ⟩− q−3

⟨ ⟩= (q1 − q−1)

⟨ ⟩⟨ ⟩

⟨ ⟩= q2 + 1 + q−2

Alternative relations⟨ ⟩= q2

⟨ ⟩− q3

⟨ ⟩,⟨ ⟩

= q−2⟨ ⟩

− q−3⟨ ⟩

Theorem (Freyd – Hoste – Lickorish – Millett – Ocneanu – Yetter,Przytycki – Traczyk)

There exists a unique Laurent polynomial invariant 〈·〉 satisfyingthe following relations:

⟨ ⟩− q−3

⟨ ⟩= (q1 − q−1)

⟨ ⟩⟨ ⟩

⟨ ⟩= q2 + 1 + q−2

Alternative relations⟨ ⟩= q2

⟨ ⟩− q3

⟨ ⟩,⟨ ⟩

= q−2⟨ ⟩

− q−3⟨ ⟩

Definition (Kuperberg, 96)

A web is a plane oriented trivalent graph with possibly somevertexless circles.

Kuperberg bracket

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

Kuperberg bracket

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

⟨ ⟩Kuperberg bracket

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

[3][2]2

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

[3][2]3

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

[3][2]3

⟨ ⟩

+[3][2]3

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

2[3][2]5⟨ ⟩

Kuperberg bracket

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

2[3]2[2]5

Kuperberg bracket

⟨ ⟩=⟨ ⟩

+⟨ ⟩

,⟨ ⟩= [2] ·

⟨ ⟩,⟨

t w⟩

= [3] · 〈w〉 .

Theorem (Khovanov, 2003)

There exists a TQFT F from the category of webs and foams tothe category of graded Z-modules which categorifies theKuperberg bracket. This TQFT allows to define the sl3-homologywhich categories the sl3 polynomial.

Theorem (Khovanov, 2003)

There exists a TQFT F from the category of webs and foams tothe category of graded Z-modules which categorifies theKuperberg bracket. This TQFT allows to define the sl3-homologywhich categories the sl3 polynomial.

Sequences of signs ε

7→ Algebras

Web-tangles w

7→ Bi-modules

+ + − −

Foams (with corners) f

7→ Bi-modules maps

Sequences of signs ε 7→ Algebras

Web-tangles w 7→ Bi-modules

+ + − −

Foams (with corners) f 7→ Bi-modules maps

First attempt:

K ε def=⊕w1,w2

F(w1w2)

Definition (Mackaay – Pan –Tubbenhauer, R.)

K ε def=

⊕w1,w2

non-elliptic

F(w1w2)

An ε-web is non-elliptic if it

contains: NO , NO , and

Theorem (Kuperberg, 96)

For a given ε, there existfinitely many non-ellipticε-webs.

First attempt:

K ε def=⊕w1,w2

F(w1w2)

K ε def=

⊕w1,w2

non-elliptic

F(w1w2)

First attempt:

K ε def=⊕w1,w2

F(w1w2)

K ε def=

⊕w1,w2

non-elliptic

F(w1w2)

fε× I ⊗

g 7→

f g ε× I

Definition (Mackaay – Pan – Tubbenhauer, R.)

F(w)def=

⊕w1∈NE(ε1)w2∈NE(ε2)

F(w1ww2) F(u)def=

∑w1∈NE(ε1)w2∈NE(ε2)

F ((w1 × I )u(w2 × I ))

afbu u(f )f

fε× I ⊗

g 7→

f g ε× I

F(w)def=

F(w1ww2) F(u)def=

F ((w1 × I )u(w2 × I ))

afbu u(f )f

fε× I ⊗

g 7→

f g ε× I

F(w)def=

F(w1ww2) F(u)def=

F ((w1 × I )u(w2 × I ))

u u(f )f

fε× I ⊗

g 7→

f g ε× I

F(w)def=

F(w1ww2) F(u)def=

F ((w1 × I )u(w2 × I ))

u u(f )f

Theorem (Mackaay – Pan – Tubbenhauer, R.)

This construction leads to a well-defined 2-functor from the2-category of web-tangles to the 2-category of algebras.

Theorem (Mackaay – Pan – Tubbenhauer, R.)

From this we can derived an extension of the sl3-homology totangles. It satisfies the following gluing relation at the level ofcomplexes:

C (T1T2) ' C (T1)⊗K ε1

C (T2).

Question

How are the projective indecomposable modules over K ε ?

Proposition

If w1 and w2 are two ε-webs, the graded dimension ofhomK ε(F(w1),F(w2)) is given by:

ql(ε) · 〈w1w2〉 .

+ − − +

q4 ·⟨ ⟩

= 1 + 3q2 + 4q4 + 3q6 + q8

Consequence

This web-module is indecomposable.

Question

Proposition

ql(ε) · 〈w1w2〉 .

+ − − +

q4 ·⟨ ⟩

= 1 + 3q2 + 4q4 + 3q6 + q8

Consequence

Question

Proposition

ql(ε) · 〈w1w2〉 .

+ − − +

q4 ·⟨ ⟩

= 1 + 3q2 + 4q4 + 3q6 + q8

Consequence

q12 〈ww〉 = 2 + 80q2 + 902q4 + 4604q6 + 13158q8 + 23684q10 +28612q12+23684q14+13158q16+4604q18+902q20+80q22+2q24

w = w0 =

Definition (R.)

An ε-web is superficial if it has no nested face.

Theorem (R.)

If w be a superficial and non-elliptic ε-web, then the K ε-moduleF(w) is indecomposable. If w ′ is another superficial andnon-elliptic ε-web different from w, then F(w) and F(w ′) are notisomorphic as K ε-modules.

Around the boundary

A semi-non-elliptic superficial ε-web is equal (in the skein module)to a sum of non-elliptic webs.

Around the boundary

A semi-non-elliptic superficial ε-web is equal (in the skein module)to a sum of non-elliptic webs.

Proposition (last section)

Let w be an ε-web.〈ww〉 is monic of degree l(ε) ⇒ F(w) is indecomposable.

Combinatorial data (red graphs) to idempotent foams,

“Non-monicity” to red graphs.

Theorem (R.)

Let w be an ε-web.〈ww〉 is monic of degree l(ε) ⇔ F(w) is indecomposable.

Theorem (R.)

Let w be an ε-web.〈ww〉 is monic of degree l(ε) ⇔ F(w) is indecomposable.

Definition (R.)

Let w be an ε-web, a red graph for G is an induced subgraph ofthe dual of w such that:

f1f2f3

⇒ {f1, f2, f3} * V (G ).

Proposition (R.)

From any exact red graph, one can construct a non-trivialidempotent.

Proposition (R.)

(Global) Exactness ! foam of degree 0.

Foam relations i ◦ p = λidw ′ .

(Local) Exactness λ 6= 0.

Enhanced red graphs to encode foams.

If 〈ww〉 is not monic of degree l(ε), then there exists an exact redgraph for w.

Idea of the proof

Back-tracking the degree 0 contribution in the computation of theKuperberg bracket.

If 〈ww〉 is not monic of degree l(ε), then there exists an exact redgraph for w.

Idea of the proof

Back-tracking the degree 0 contribution in the computation of theKuperberg bracket.

Web colourings to give a new definition of TQFT ?

Web colourings with a geometrical meaning.

Commutation of idempotents.

There exists other kinds of webs: so(5), so(2n).

Sur l'homologie sl3 des enchevêtrements ; algèbres de ... · Introduction Algebras K Super...

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