categorifying higher su3 knot...

Categorifying higher su3 knot polynomials

David [email protected]

Randolph-Macon CollegeAshland, VA

University of VirginiaTopology Seminar

March 29, 2011

David Clark Categorifying higher su3 knot polynomials

The quantum su3 link polynomial

Using the skein relations,

7−→ q2 − q3

7−→ −q−3 + q−2

subject to Kuperberg’s su3 spider relations,

= q2 + 1 + q−2 = q + q−1

Using the skein relations,

7−→ q2 − q3

7−→ −q−3 + q−2

subject to Kuperberg’s su3 spider relations,

= q2 + 1 + q−2 = q + q−1

. . . we get an assignment

L 7−→ J su3(L),

a specialization of the HOMFLY polynomial.

From a representation theoretic standpoint, this polynomialcomes from coloring the link with the fundamental vectorrepresentation V ∼= C3.

. . . we get an assignment

L 7−→ J su3(L),

a specialization of the HOMFLY polynomial.

From a representation theoretic standpoint, this polynomialcomes from coloring the link with the fundamental vectorrepresentation V ∼= C3.

Original categorification

Khovanov categorified this polynomial

Theorem (Khovanov)

χ(Kh(L)) = Jsu3(L)

L1 L1Kh( )

Theorem (Khovanov)

χ(Kh(L)) = Jsu3(L)

L1 L1Kh( )

Theorem (Khovanov)

χ(Kh(L)) = Jsu3(L)

Theorem (Khovanov)

χ(Kh(L)) = Jsu3(L)

Theorem (Khovanov)

χ(Kh(L)) = Jsu3(L)

Theorem (Khovanov)

χ(Kh(L)) = Jsu3(L)

“Algebra Light” categorification

Morrison and Nieh gave a “universal” categorification of thisinvariant, allowing us to linger in the realm of pictures a bitlonger.

Maps are now cobordisms between webs, called “foams.”

Categorified skein relations:

� //

(• // q2 // q3 // •

� //

(• // q−3 // q−2 // •

� //

(• // q2 // q3 // •

� //

(• // q−3 // q−2 // •

� //

(• // q2 // q3 // •

� //

(• // q−3 // q−2 // •

Categorified spider relations (over Q)

+ + = 0

= 12 + 1

= 13 − 1

9 + 13 = −

Useful properties

This view of Khovanov’s su3 theory is

“universal,” in that it’s independent of the chosenalgebraic formulation.“local,” in that it’s built with tangles in mind.“easy,” because it’s completely combinatorial.

Useful properties

“universal,” in that it’s independent of the chosenalgebraic formulation.

“local,” in that it’s built with tangles in mind.“easy,” because it’s completely combinatorial.

Useful properties

“universal,” in that it’s independent of the chosenalgebraic formulation.“local,” in that it’s built with tangles in mind.

“easy,” because it’s completely combinatorial.

Useful properties

“universal,” in that it’s independent of the chosenalgebraic formulation.“local,” in that it’s built with tangles in mind.“easy,” because it’s completely combinatorial.

Useful properties

Theorem (C.)

The su3 Khovanov homology is properly functorial with respect tolink cobordisms, i.e.,

Σ ' Σ′ ⇒ Kh(Σ) = Kh(Σ′)

Functoriality allows us to explore the su3 link homology inmore subtle ways . . .

Useful properties

Theorem (C.)

Σ ' Σ′ ⇒ Kh(Σ) = Kh(Σ′)

Useful properties

Theorem (C.)

Σ ' Σ′ ⇒ Kh(Σ) = Kh(Σ′)

Bigger picture

The homology theory we’ve been discussing categorifies thepolynomial corresp to Vfund = C3.

Jsu3(K ) = Jsu3fund(K )

But there are polynomials obtained by coloring a link with anyirrep Vλof su3.

Jsu3λ (K )

Ben Webster has categorified these invariants in analgebro-geometric setting.

Bigger picture

Jsu3λ (K )

Bigger picture

Jsu3λ (K )

Our goal

Our goal: to categorify these higher su3 polynomials in thislocal, combinatorial setting.

Possible strategies:

Categorify the su3 Jones-Wenzl idempotents.Use representation theory, and work with the symmetricgroup.

Our goal

Possible strategies:Categorify the su3 Jones-Wenzl idempotents.

Use representation theory, and work with the symmetricgroup.

Our goal

Possible strategies:Categorify the su3 Jones-Wenzl idempotents.Use representation theory, and work with the symmetricgroup.

An action of the symmetric group

Fix a knot K, and consider its n-parallel cable:

K K(n)

Let Ri : K (n) → K (n) be the cobordism that swaps the ith and(i + 1)st cables via the right-hand rule.

Any composition of such cobordisms induces a map on theKhovanov homology of the n-cable:

Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))

So let Sn act on Kh(Kn) via these maps!

Theorem (C.)

This is an honest group action, i.e., the map

Sn −→ End(Kh(K (n)))

σi 7−→ Kh(Ri )

is a homomorphism of groups.

So let Sn act on Kh(Kn) via these maps!

Theorem (C.)

This is an honest group action, i.e., the map

Sn −→ End(Kh(K (n)))

σi 7−→ Kh(Ri )

is a homomorphism of groups.

Sketch of proof

Sketch of proof.

Our action needs to satisfy the relations on transpositions in Sn:

1 σiσj = σjσi if j 6= i ± 1

2 σiσi+1σi = σi+1σiσi+1

3 σ2i = 1

For relations (1) and (2), we need to show that

Kh(RiRj) = Kh(RjRi ) if j 6= i ± 1

Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1)

Sketch of proof

Sketch of proof.

3 σ2i = 1

Sketch of proof

Sketch of proof.

3 σ2i = 1

Sketch of proof

Conveniently, these both follow directly from functoriality!

Relation (1):

Functoriality⇒ Kh(RiRj) = Kh(RjRi ).

Sketch of proof

Relation (1):

Sketch of proof

Relation (1):

Sketch of proof

Relation (1):

R3 1R R31R

Sketch of proof

Relation (1):

R3 1R R31R

Sketch of proof

Relation (1):

R3 1R R31R

Sketch of proof

Relation (2):

Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).

Sketch of proof

Relation (2):

1R 2R 1R

Sketch of proof

Relation (2):

1R 1R2R 1R 2R 2R

Sketch of proof

Relation (2):

1R 1R2R 1R 2R 2R

Sketch of proof

Relation (2):

1R 1R2R 1R 2R 2R

Sketch of proof

. . . but for relation (3), we need to show that

Kh(R2i ) = Id

So we need to look more carefully at the induced maps . . .

Sketch of proof

Kh(R2i ) = Id

Sketch of proof

Kh(R2i ) = Id

1R Id2

Sketch of proof

Kh(R2i ) = Id

1R Id2

Sketch of proof

Kh(R2i ) = Id

1R Id2

Sketch of proof

Mercifully, it will suffice to consider the 2-cable of K .

In particular, we need a movie of knot diagrams that describesthe cobordism R .

Sketch of proof

Mercifully, it will suffice to consider the 2-cable of K .

In particular, we need a movie of knot diagrams that describesthe cobordism R .

Sketch of proof

T Tx Tx

This is a pair of R2 moves on the ends, with 4c R3 movesin the middle (where c is the number of crossing in theoriginal knot K ).That’s a very nasty map to compute explicitly!

Sketch of proof

T Tx Tx

This is a pair of R2 moves on the ends, with 4c R3 movesin the middle (where c is the number of crossing in theoriginal knot K ).

That’s a very nasty map to compute explicitly!

Sketch of proof

T Tx Tx

This is a pair of R2 moves on the ends, with 4c R3 movesin the middle (where c is the number of crossing in theoriginal knot K ).That’s a very nasty map to compute explicitly!

Sketch of proof

Instead, consider the cobordism L:

Sketch of proof

However, with some work 1 one can show that

Kh(L) = Kh(R)

And notice that

1using the Categorified Kauffman Trick, and other tricks.David Clark Categorifying higher su3 knot polynomials

Sketch of proof

Kh(L) = Kh(R)

And notice that

Sketch of proof

Kh(L) = Kh(R)

And notice that

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(RiRj) = Kh(RjRi ) X

Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X

Kh(R2i ) = Id X

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(R2i ) = Id X

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(R2i ) = Id X

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(R2i ) = Id X

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(R2i ) = Id X

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(R2i ) = Id X

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(R2i ) = Id X

Sketch of proof

Thus, we see that

Kh(R2) = Kh(R · R)

= Kh(L · R)

= Kh(Id)

Kh(R2i ) = Id X

So ...

Why do we care?

More categorification

Recall: our goal is to categorify the polynomials Jsu3λ :

K ,Vλ Jsu3λ (K )

Basic idea:

1 For a knot K , we’ll find the (huge!) Khovanov complex ofone of its parallel cables.

2 Using our symmetric group action, we’ll project down to acomplex whose Euler characteristic is Jsu3λ (K ).

K ,Vλ Jsu3λ (K )

Basic idea:

1 For a knot K , we’ll find the (huge!) Khovanov complex ofone of its parallel cables.

K ,Vλ Jsu3λ (K )

Basic idea:1 For a knot K , we’ll find the (huge!) Khovanov complex of

one of its parallel cables.

K ,Vλ Jsu3λ (K )

Basic idea:1 For a knot K , we’ll find the (huge!) Khovanov complex of

one of its parallel cables.2 Using our symmetric group action, we’ll project down to a

complex whose Euler characteristic is Jsu3λ (K ).

Some representation theory

Let V = C3 be the standard vector representation of su3.

Recall: there is a two-parameter family of irreps Vλ of su3,parameterized by λ = (λ1, λ2) ∈ Z2

Fact: for n = λ1 + 2λ2, we know that Vλ is asubrepresentation of V⊗n.

There is an idempotent sλ ∈ End(V⊗n) that projects ontoVλ.

The map sλ, sometimes called the Schur functor, is reallyjust a linear combination of permutations of the tensorpowers of V⊗n, and can thus be viewed as

sλ ∈ QSn.

For example, to get the adjoint representation Vad, we canproject

sad : V ⊗ V ⊗ V −→ Vad

by letting

sad =1

(Id + τ(1 2) − τ(1 3) − τ(1 3 2)

For example, to get the adjoint representation Vad, we canproject

sad : V ⊗ V ⊗ V −→ Vad

by letting

sad =1

(Id + τ(1 2) − τ(1 3) − τ(1 3 2)

Proposed categorification for higher irreps

Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).

Claim: χ(Khλ(K )) = Jsu3λ (K )

��

Kh(K (n))

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Kh(K (n))

��

Vλ “Khλ(K )”

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Kh(K (n))

��

Vλ “Khλ(K )”

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Kh(K (n))

��

Vλ “Khλ(K )”

categorifying higher su3 knot...

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