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 Categorification in Knot Theory 1 Chain complex associated with sl ( N ) state model 2 The nodal foams 2-functor 3 sl ( N )-foams TQFT 4

Categorification in Knot Theory sl ( N ) state model The nodal foams 2-functor sl ( N )-foams TQFT Khovanov homology Theorem [Khovanov] : For each link diagram D , there exists a bigraded complex 1 K ( D ). For each Reidemeister move D ↔ D ′ there exists an homotopy 2 equivalence K ( D ) → K ( D ′ ). The graded Euler characteristic is the Jones polynomial of the 3 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 objects 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 our 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 Categorification in Knot Theory 1 Chain complex associated with sl ( N ) state model 2 The nodal foams 2-functor 3 sl ( N )-foams TQFT 4

Categorification in Knot Theory sl ( N ) state model The nodal foams 2-functor sl ( N )-foams TQFT sl ( N ) invariant of links q N P N ( ) − q − N P N ( ) = ( q − q − 1 ) P N ( ) q N − q − N P N ( ) = [ 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) P N ( G ) ∈ Z + [ q ± 1 ]. ) = q − N +1 P N ( ) − q − N P N ( P N ( ) ) = q N − 1 P N ( ) − q N P N ( P N ( )

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). D s 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. � ( − 1) s ( D ) q (1 − N ) w ( D ) − s ( D ) P N ( D s ) P N ( D ) = s

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 ) = V ( D s ) s 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 d s = � c | s ( c ) | free abelian group generated by crossings c with | s ( c ) | = 1 (or double edges of D s ), equipped with standard bilinear form. � V ( D s ) ⊗ ∧ d s ∆ s K ( D ) = s For a positive crossing c : δ = V (Σ) ⊗ ( • ∧ c ) : V ( D s ) ⊗ ∧ d s ∆ s → V ( D s ′ ) ⊗ ∧ d s ′ ∆ s ′ For a negative crossing c , δ = V (Σ ′ ) ⊗ < • , c > : V ( D s ) ⊗ ∧ d s ∆ s → V ( D s ′ ) ⊗ ∧ d s ′ ∆ 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 Categorification in Knot Theory 1 Chain complex associated with sl ( N ) state model 2 The nodal foams 2-functor 3 sl ( N )-foams TQFT 4

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