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An introduction to link homology Marco Mackaay CAMGSD and - - PowerPoint PPT Presentation

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History Link polynomials Link homologies An introduction to link homology Marco Mackaay CAMGSD and Universidade do Algarve 2 September, 2013 1/39 Marco Mackaay An introduction to link homology History Link polynomials Link homologies

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History Link polynomials Link homologies

An introduction to link homology

Marco Mackaay

CAMGSD and Universidade do Algarve

2 September, 2013

1/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Outline

1

History

2/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Outline

1

History

2

Link polynomials

2/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Outline

1

History

2

Link polynomials

3

Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

2/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Section outline

1

History

2

Link polynomials

3

Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

3/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Link polynomials and homologies

L − → P(L) polynomial ( Jones ’84 + HOMFLY-PT ’85)

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History Link polynomials Link homologies

Link polynomials and homologies

L − → P(L) polynomial ( Jones ’84 + HOMFLY-PT ’85) L − → H(L) homology ( Khovanov ’99 + Khovanov-Rozansky ’04 )

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History Link polynomials Link homologies

Link polynomials and homologies

L − → P(L) polynomial ( Jones ’84 + HOMFLY-PT ’85) L − → H(L) homology ( Khovanov ’99 + Khovanov-Rozansky ’04 ) χq

  • H(L)
  • = P(L)

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History Link polynomials Link homologies

Short history of foams

1

sl(2) link homology Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010)

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History Link polynomials Link homologies

Short history of foams

1

sl(2) link homology Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010)

5/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Short history of foams

1

sl(2) link homology Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010)

2

sl(3) link homologies using sl(3)-foams Khovanov (2004), M.-Vaz (2007), Morrisson-Nieh (2008), Lauda-Queffelec-Rose (2012)

5/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Short history of foams

1

sl(2) link homology Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010)

2

sl(3) link homologies using sl(3)-foams Khovanov (2004), M.-Vaz (2007), Morrisson-Nieh (2008), Lauda-Queffelec-Rose (2012)

3

sl(N) (N > 3) link homologies using foams Khovanov-Rozansky (2004), M.-Stoˇ si´ c-Vaz (2007).

5/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Section outline

1

History

2

Link polynomials

3

Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

6/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies

Reidemeister moves

R1 R2 R3

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History Link polynomials Link homologies

The sl(N)-link polynomial

Resolutions in MOY-webs (Murakami-Ohtsuki-Yamada)

1

  • 1
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History Link polynomials Link homologies

= [N],

  • =

N 2

  • ,

Γ⊔Γ′ = ΓΓ′ = [2] = [N −1] = +[N −2] + = +

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History Link polynomials Link homologies

The Jones polynomial of the trefoil (N = 2)

1 3 2

q(q+q−1)

100

+ q2(q+q−1)2

110

  • +

(q+q−1)2

000

  • q(q+q−1)

010

  • +

q2(q+q−1)2

101

+ q3(q+q−1)3

111

  • q(q+q−1)

001

  • q2(q+q−1)2

011

  • (q+q−1)2

− 3q(q+q−1) + 3q2(q+q−1)2 − q3(q+q−1)3 = q−2+1+q2−q6

·(−1)n− qn+−2n− (q+q−1)−1

− − − − − − − − − − − − − − − →

(with (n+,n−) = (3,0))

J(T)=q2+q6−q8.

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Section outline

1

History

2

Link polynomials

3

Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Khovanov link homology

The main ideas Let A = Q[X]/(X2), such that deg(1) = 0 and deg(X) = 2. So qdim(A{−1}) = q+q−1. Associate to each resolution with s circles the vector space A⊗s{t}. Take direct sum over each column. Resolutions in consecutive columns can be obtained from

  • ne another by merging two circles into one or splitting one

circle into two. Define a coproduct on A by 1 → 1⊗X +X ⊗1 X → X ⊗X. Use the product and coproduct in A to define differentials.

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Khovanov homology of the trefoil

:

111

00* 0*0 *00 *01 0*1 *10 1*0 *11 1*1 10* 01* 11*

1− 2− 3− 000 001 010 100 011 101 110

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Lemma A is a Frobenius algebra with trace ε(1) = 0 and ε(X) = 1.

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Lemma A is a Frobenius algebra with trace ε(1) = 0 and ε(X) = 1. Lemma A defines a 2-dimensional TQFT, i.e. a monoidal functor Oriented surfaces → Vect.

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Lemma A is a Frobenius algebra with trace ε(1) = 0 and ε(X) = 1. Lemma A defines a 2-dimensional TQFT, i.e. a monoidal functor Oriented surfaces → Vect. Corollary We have d2 = 0.

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Theorem (Khovanov) Khovanov’s complex is invariant up to homotopy equivalence under the Reidemeister moves. Thus the homology is a link invariant.

15/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Theorem (Khovanov) Khovanov’s complex is invariant up to homotopy equivalence under the Reidemeister moves. Thus the homology is a link invariant. Question (Bar-Natan) What is the kernel of Khovanov’s TQFT?

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Bar-Natan’s cobordisms

Theorem (Bar-Natan) The kernel of Khovanov’s TQFT is generated by: (2D): = 0 (S): = 0, = 1 (NC): =

  • +
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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

The first Reidemeister move

  • :
  • :

g0 =

  • d =
  • h =
  • f 0 =

1−i i

−

1

∑

i=0

(−1)i

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Recovering Khovanov homology from Bar-Natan’s cobordisms

Let Cob/ℓ be the category of cobordisms modulo the Bar-Natan relations. A cobordism f has degree q( f) = −χ( f)+2d.

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Recovering Khovanov homology from Bar-Natan’s cobordisms

Let Cob/ℓ be the category of cobordisms modulo the Bar-Natan relations. A cobordism f has degree q( f) = −χ( f)+2d. Γ complete resolution: F(Γ) = HomCob/ℓ(/ 0,Γ), F(Γ) graded

18/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Recovering Khovanov homology from Bar-Natan’s cobordisms

Let Cob/ℓ be the category of cobordisms modulo the Bar-Natan relations. A cobordism f has degree q( f) = −χ( f)+2d. Γ complete resolution: F(Γ) = HomCob/ℓ(/ 0,Γ), F(Γ) graded f : Γ → Γ′ cobordism: F( f) : HomCob/ℓ(/ 0,Γ) → HomCob/ℓ(/ 0,Γ′) given by composition, whose degree equals q( f).

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sl(3)-Foams and link homology

Recall:

  • 1
  • 1
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sl(3)-Foams and link homology

Recall:

  • 1
  • 1
  • The differential:

− − − − − − − − − − →

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Khovanov’s local relations: ℓ = (3D,CN,S,Θ)

= 0 (1) = − − − (2) = = 0, = −1 (3) =    1 (α,β,γ) = (1,2,0) or a cyclic permutation −1 (α,β,γ) = (2,1,0) or a cyclic permutation else (4) ℓ suffice to evaluate any closed foam!

20/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Definition For any webs Γ and Γ′ there is a closure map Foam(/ 0,Γ)⊗Foam(Γ,Γ′)⊗Foam(Γ, / 0) → Foam(/ 0, / 0) ∼ = Q. Define Foam/ℓ to be the category whose objects are webs and let Hom/ℓ(Γ,Γ′) be the vector space of foams modulo the foams for which all closures evaluate to zero.

21/39 Marco Mackaay An introduction to link homology

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Theorem (Khovanov) To each link diagram L one can associate a homology complex L3 which is invariant up homotopy equivalence under the Reidemeister moves. Furthermore χq(L3) = P

3(L).

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Universal sl(3)-link homology

= a +b +c (5) = − − − +a

  • +
  • +b

(6)

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

Universal sl(3)-link homology

= a +b +c (5) = − − − +a

  • +
  • +b

(6) Theorem (M.M.-Vaz) The relations (3), (4), (5) and (6) still give rise to an invariant link homology There are three isomorphism classes, depending on the nr.

  • f distinct roots of f(x) = x3 +ax2 +bx+c

See also work by Morrison and Nieh (2008).

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

The sl(N)-link homology (N > 3)

Still the same:

  • 1
  • 1
  • And so is:

− − − − − − − − − − →

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History Link polynomials Link homologies Khovanov link homology sl(3) link homology sl(N) link homology

sl(N)-foams

Facets: , , Edges: , Vertices:

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Dots: · Simple facet

  • ℄

(X) · Double facet

  • ℄,
^

(π1,0,π1,1) · Triple facet

  • ℄,
^, _

(π1,0,0,π1,1,0,π1,1,1)

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Dots: · Simple facet

  • ℄

(X) · Double facet

  • ℄,
^

(π1,0,π1,1) · Triple facet

  • ℄,
^, _

(π1,0,0,π1,1,0,π1,1,1) Let πλ be a Schur polynomial. Define

i

,

(k,m)

and

* (p,q,r)

using the Jacobi-Trudi formula,

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Dots: · Simple facet

  • ℄

(X) · Double facet

  • ℄,
^

(π1,0,π1,1) · Triple facet

  • ℄,
^, _

(π1,0,0,π1,1,0,π1,1,1) Let πλ be a Schur polynomial. Define

i

,

(k,m)

and

* (p,q,r)

using the Jacobi-Trudi formula, e.g.:

(3,0)

= −2

  • π3,0 = π3

1,0 −2π1,0π1,1

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Evaluation of sl(N)-foams

Kapustin-Li, Khovanov-Rozansky, M.-Stoˇ si´ c-Vaz To evaluate closed foams use the Kapustin-Li formula

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Evaluation of sl(N)-foams

Kapustin-Li, Khovanov-Rozansky, M.-Stoˇ si´ c-Vaz To evaluate closed foams use the Kapustin-Li formula Definition FoamN is the category of webs and foams modulo the kernel of the KL-evaluation Objects: Closed webs Morphisms: Q-Linear combinations of isotopy classes of foams modulo the relation:

∑ci fi = 0

if ∑ci ¯ fi evaluates to zero for all closures.

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Relations in FoamN

Theorem (M.-Stoˇ si´ c-Vaz) The following relations hold in FoamN: (Dot Conversion)

i

= 0, i ≥ N

(k,m)

= 0, k ≥ N −1

* (p,q,r)

= 0, p ≥ N −2

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(Dot Migration) = + =

*

=

*

+

* *

=

*

+

* *

=

*

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(Matveev-Piergalini)

*

=

*

,

* = *

(MP)

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(Matveev-Piergalini)

*

=

*

,

* = *

(MP) (Neck cutting) =

N−1

∑

i=0

N−1−i i

(NC1)

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(Matveev-Piergalini)

*

=

*

,

* = *

(MP) (Neck cutting) =

N−1

∑

i=0

N−1−i i

(NC1) =

∑

0≤ j≤i≤N−2

(i,j) (i,j)

(NC2)

* =

∑

0≤k≤ j≤i≤N−3

* *

(i,j,k) (i,j,k)

(NC3)

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(Spheres)

i

=

  • 1,

i = N −1 0, else (S1)

(i,j)

=

  • −1,

i = j = N −2 0, else (S2)

*

(i,j,k) =

  • −1,

i = j = k = N −3 0, else (S3)

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(Theta foams)

N−1 N−2 = 1 = − N−2 N−1

( C1)

(N−3,N−3,N−3)

= 1 = −

(N−3,N−3,N−3)

*

( C2)

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(Theta foams)

N−1 N−2 = 1 = − N−2 N−1

( C1)

(N−3,N−3,N−3)

= 1 = −

(N−3,N−3,N−3)

*

( C2) (MOY relations) = − (DR1) =

∑

a+b+c=N−2

b a c

(DR2) + other digon relations ...

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= − +

∑

a+b+c+d=N−3

c b a d

(SqR1)

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= − +

∑

a+b+c+d=N−3

c b a d

(SqR1) and a special relation: = − +

*

(SqR2)

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Theorem (M.-Stoˇ si´ c-Vaz) is invariant up to homotopy equivalence under the Reidemeister moves.

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Theorem (M.-Stoˇ si´ c-Vaz) is invariant up to homotopy equivalence under the Reidemeister moves. Theorem (M.-Stoˇ si´ c-Vaz) is isomorphic to the Khovanov-Rozansky homology complex.

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Reidemeister I: D = D′ = D ≃ D′ D : D′ :

g0 =

  • d =
  • h =

a c b a+b+c=N−2

Σ

  • f 0 =

N−1

∑

i=0

  • E.g.

d f 0 = 0 by Dot migration & Dot conversion

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Reidemeister III: D = D′ =

D : : Q

− − −I I −I −I I I

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Theorem (M.-Stoˇ si´ c-Vaz) is functorial under link cobordisms up to scalars.

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Theorem (M.-Stoˇ si´ c-Vaz) is functorial under link cobordisms up to scalars. Conjecture Functoriality holds on the nose. sl(2) - modified knot homologies of Clark, Morrison and Walker (2007), Caprau (2007) and Blanchet (2010) sl(3) - Clark (2009) sl(N) (N > 3) - ?

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An example: T = H0(T) ∼ =

  • Q[x]/xN

{N −1} H1(T) = 0 H2(T) ∼ =

N−2

  • i=0

Q{3(N −1)+2(1−i)} H3(T) ∼ =

N−2

  • i=0

Q{3(N −1)+2(1−i)+2N} PN(T) = q2(N−1)[N]+t2q2N+1[N −1]+t3q4N+1[N −1]

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THANKS!!!

39/39 Marco Mackaay An introduction to link homology