Nodal foams and link homology Christian Blanchet IMJ, Universit e - - PowerPoint PPT Presentation

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Nodal foams and link homology

Christian Blanchet

IMJ, Universit´ e Paris-Diderot

Zurich - December 11, 2009

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

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Categorification in Knot Theory

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Chain complex associated with sl(N) state model

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The nodal foams 2-functor

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sl(N)-foams TQFT

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Khovanov homology

Theorem [Khovanov] :

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For each link diagram D, there exists a bigraded complex K(D).

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For each Reidemeister move D ↔ D′ there exists an homotopy equivalence K(D) → K(D′).

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The graded Euler characteristic is the Jones polynomial of the link.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Khovanov homology functor

Diagram of a link cobordism (movie) → chain map. Projective functoriality (Khovanov, Jacobsson, Bar-Natan). For original Khovanov homology, the map induced on homology is well defined up to sign. Clark-Morrisson-Walker, Caprau. Strictly functorial sl(2) link homology over Z[i]. B. Strictly functorial sl(2) link homology over Z.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

The tangle category

A tangle is a link with boundary in [0, 1] × C. Tangles form a category in which

• bjects are finite sequences of signs interpreted as sets of

standard oriented points in the real line, morphisms are tangles with boundary prescribed by source in {0} × C and target in {1} × C.

In the tangle 2-category, 2-morphisms are cobordisms embedded in [0, 1] × [0, 1] × C, up to isotopy.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Khovanov homology for tangles

1 Bar-Natan has constructed a (graded) 2-functor from the

tangle 2-category to a topological (or diagrammatic) 2-category).

2 Khovanov homology is obtained by composing the above

2-functor with a TQFT.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Categorification of sl(N) link invariant

Khovanov-Rozansky homology categorifies sl(N) quantum invariant and Homfly quantum invariant (over Q). (Jones is sl(2) ; sl(N), N ≥ 2, are specializations of Homfly.) Khovanov : topological construction of a categorification of sl(3) quantum invariant. Mackaay-Stosic-Vaz : topological categorification of sl(N) quantum invariant using foams and Kapustin-Li formula (over Q). Sussan, Mazorchuk-Stroppel : sl(N) link homology from category O. Khovanov : Homfly link homology using Soergel bimodules.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Results

We define a 2-category whose 1-morphisms are trivalent graphs, and 2-morphisms are nodal foams up to relations. We obtain a 2-functor (` a la Bar Natan) from the tangle 2-category to the homotopy category of formal complexes in

• ur nodal foams category ; we call it the nodal foams

2-functor. For each N ≥ 2 we give an alternative construction of Mackay-Stosik-Vaz sl(N)-foams TQFT using cohomology of partial flag manifolds, and obtain sl(N) link homology by composing the nodal foams functor with sl(N)-foams TQFT.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

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Categorification in Knot Theory

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Chain complex associated with sl(N) state model

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The nodal foams 2-functor

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sl(N)-foams TQFT

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

sl(N) invariant of links

qN PN( ) − q−N PN( ) = (q − q−1) PN( ) PN( ) = [N] =

qN−q−N q−q−1

Jones polynomial : N = 2

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

State sum for sl(N), local formula

The sl(N) invariant can be obtained from an invariant of planar graphs (Murakami-Ohtsuki-Yamada) PN(G) ∈ Z+[q±1]. PN( ) = q−N+1 PN( ) − q−N PN( ) PN( ) = qN−1 PN( ) − qN PN( )

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

State sum for sl(N), global formula

A state s of a link diagram D associates to a positive (resp. negative) crossing either 0 or 1 (resp. −1 or 0). Ds is a planar trivalent graph, defined by the rule : if s(c) = 0, then c is replaced by if |s(c)| = 1, then c is replaced by s(D) =

c s(c), w(D) is the writhe.

PN(D) =

• s

(−1)s(D)q(1−N)w(D)−s(D)PN(Ds)

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Building a complex, first tentative

Suppose that we have a functor : trivalent graph G → module V(G) . cobordism Σ → linear map V(Σ) . Here Σ may be a trivalent 2-complex with 2 kinds of faces (labelled respectively by 1 or 2) and singular locus (trivalent binding, singular vertices). K(D) =

• s

V(Ds) The cohomological degre is s(D) =

c s(c).

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Boundary map, first tentative

The boundary operator between summands indexed by states s and s′ is zero unless s and s′ are different only in one crossing c, where s′(c) = s(c) + 1. It is then defined using the TQFT map associated with the cobordism Σ, (resp. Σ′) which are identity outside a neighbourhood of the crossing, and are given by the saddle with membrane below, around the crossing c with s(c) = 0, s′(c) = 1 (resp. s(c) = −1, s′(c) = 0). Σ : Σ′ :

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Do we get a complex ?

By functoriality of V, all squares commute. With the above boundary map ∂, (K(D) ⊗ Z/2, ∂) is a chain complex.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Twisting the complex

Let ∆s be the rank ds =

c |s(c)| free abelian group

generated by crossings c with |s(c)| = 1 (or double edges of Ds), equipped with standard bilinear form. K(D) =

• s

V(Ds) ⊗ ∧ds∆s For a positive crossing c : δ = V(Σ) ⊗ (• ∧ c) : V(Ds) ⊗ ∧ds∆s → V(Ds′) ⊗ ∧ds′∆s′ For a negative crossing c, δ = V(Σ′)⊗ < •, c >: V(Ds) ⊗ ∧ds∆s → V(Ds′) ⊗ ∧ds′∆s′ Here < •, c > is (the antisymmetrization of) the contraction (using the standard scalar product we understand c as a form). With the above boundary map (K(D), δ) is a chain complex.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

What do we need ?

A TQFT functor V. An homotopy associated with each Reidemeister move. A chain map associated with each movie description of a cobordism satisfying a list of movie moves.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

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Categorification in Knot Theory

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Chain complex associated with sl(N) state model

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The nodal foams 2-functor

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sl(N)-foams TQFT

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

The nodal foams 2-category

The 0-objects are finite sequences of signed and 1 or 2 colored points in the real line. The 1-morphisms are trivalents graphs with flow equal to 1 or 2 on edges. The 2-morphisms are linear combinations of 2-complexes with regular faces colored 1, 2 or 3, trivalent binding, singular vertices whose link is a tetrahedron, and nodes whose neighbourhood is a cone on 2, 3 or 4 circles, up to a list of relations. We allow direct sum of 1-morphisms. We get an additive 2-category denote by F.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Bubble relations

= = 0 = 0

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Bigone reduction relations

= = 0

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Tube relations

= − =

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Surgery and four term relations

= + + = +

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Square relations

= − = +

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Node vanishing and MP relations

= 0 =

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Complexes in the nodal foams 2-category

Construction of the previous section. For each diagram of a tangle we get a complex (a 1-morphism in Komp(F)). Theorem. The homotopy class of complexes associated with a tangle is well defined. We get a 2-functor from the tangle 2-category to Komp(F)/h

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

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Categorification in Knot Theory

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Chain complex associated with sl(N) state model

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The nodal foams 2-functor

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sl(N)-foams TQFT

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

TQFT for surfaces

A TQFT in dimension 2 is a functor : curve Γ → module V(Γ) . cobordism Σ → linear map V(Σ) . TQFTs in dimension 2 ← → commutative Frobenius algebras. Extension of TQFT functor : the surface may have points labelled with an element of the Frobenius algebra.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

Example of Frobenius algebras

Cohomology of flag manifolds : A1 = A1(N) = H∗(CPN−1), A2 = A2(N) = H∗(G2(CN)) (grassmanian of planes in CN), A1,2 = A1,2(N) = H∗(G1,2(CN)) (flags lines ⊂ plane in CN), A1,2,3 = A1,2,3(N) = H∗(G1,2,3(CN)) . . . Algebra morphisms : α : A1 ⊗ A1 → A1,2 , β : A2 → A1,2 .

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

sl(N)-foams category

Objects are oriented trivalent graphs with flow equal to 1 or 2

• n edges.

Morphisms are certain compact 2-complexes :

regular faces are oriented and labelled 1, 2 or 3 ; the 1 dimensional singular locus is a trivalent binding ; singular vertices whose link is a tetrahedron.

On each trivalent vertex or binding we fix an order or the two edges or pages with the same orientation. Faces colored i may have points colored with elements of the corresponding Frobenius algebra Ai.

Categorification in Knot Theory sl(N) state model The nodal foams 2-functor sl(N)-foams TQFT

sl(N)-foams TQFT

One start with an invariant of closed foams using a state sum formula. A universal construction extends the invariant of closed foams to a functor. The sl(N)-foams TQFT extends to a 2-functor on the nodal foams category. Everything is graded.