Categorifying higher su3 knot polynomials
David Clark
[email protected]
Randolph-Macon College
Ashland, VA
University of Virginia
Topology Seminar
March 29, 2011
David Clark
Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
Using the skein relations,
q2
− q3
7−→ −q −3
+ q −2
7−→
David Clark
Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
Using the skein relations,
q2
− q3
7−→ −q −3
+ q −2
7−→
subject to Kuperberg’s su3 spider relations,
= q 2 + 1 + q −2
=
David Clark
=q
+ q −1
+
Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
. . . we get an assignment
L 7−→ J su3 (L),
a specialization of the HOMFLY polynomial.
David Clark
Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
. . . we get an assignment
L 7−→ J su3 (L),
a specialization of the HOMFLY polynomial.
From a representation theoretic standpoint, this polynomial
comes from coloring the link with the fundamental vector
representation V ∼
= C3 .
David Clark
Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
David Clark
Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
David Clark
Kh( L 1 )
Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
Kh( L 1 )
Theorem (Khovanov)
χ(Kh(L)) = J su3 (L)
David Clark
Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
Kh( L 1 )
Kh( L 2 )
L2
Theorem (Khovanov)
χ(Kh(L)) = J su3 (L)
David Clark
Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
Kh( L 1 )
Σ
Kh( L 2 )
L2
Theorem (Khovanov)
χ(Kh(L)) = J su3 (L)
David Clark
Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
Kh( L 1 )
Σ
Kh( Σ )
Kh( L 2 )
L2
Theorem (Khovanov)
χ(Kh(L)) = J su3 (L)
David Clark
Categorifying higher su3 knot polynomials
“Algebra Light” categorification
Morrison and Nieh gave a “universal” categorification of this
invariant, allowing us to linger in the realm of pictures a bit
longer.
David Clark
Categorifying higher su3 knot polynomials
“Algebra Light” categorification
Morrison and Nieh gave a “universal” categorification of this
invariant, allowing us to linger in the realm of pictures a bit
longer.
Maps are now cobordisms between webs, called “foams.”
David Clark
Categorifying higher su3 knot polynomials
“Algebra Light” categorification
Morrison and Nieh gave a “universal” categorification of this
invariant, allowing us to linger in the realm of pictures a bit
longer.
Maps are now cobordisms between webs, called “foams.”
Categorified skein relations:
/
/
•
•
/ q −3
David Clark
/ q2
/ q −2
!
/ q3
/•
/•
!
Categorifying higher su3 knot polynomials
Categorified spider relations (over Q)
=0
+
+
=0
=3
=0
=
1
2
+
1
2
=0
=
1
3
−
1
9
+
1
3
=−
David Clark
Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
David Clark
Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
“universal,” in that it’s independent of the chosen
algebraic formulation.
David Clark
Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
“universal,” in that it’s independent of the chosen
algebraic formulation.
“local,” in that it’s built with tangles in mind.
David Clark
Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
“universal,” in that it’s independent of the chosen
algebraic formulation.
“local,” in that it’s built with tangles in mind.
“easy,” because it’s completely combinatorial.
David Clark
Categorifying higher su3 knot polynomials
Useful properties
L1
Kh( L 1 )
Σ
Kh( Σ )
Kh( L 2 )
L2
David Clark
Categorifying higher su3 knot polynomials
Useful properties
L1
Kh( L 1 )
Σ
Kh( Σ )
Kh( L 2 )
L2
Theorem (C.)
The su3 Khovanov homology is properly functorial with respect to
link cobordisms, i.e.,
Σ ' Σ0 ⇒ Kh(Σ) = Kh(Σ0 )
David Clark
Categorifying higher su3 knot polynomials
Useful properties
L1
Kh( L 1 )
Σ
Kh( Σ )
Kh( L 2 )
L2
Theorem (C.)
The su3 Khovanov homology is properly functorial with respect to
link cobordisms, i.e.,
Σ ' Σ0 ⇒ Kh(Σ) = Kh(Σ0 )
Functoriality allows us to explore the su3 link homology in
more subtle ways . . .
David Clark
Categorifying higher su3 knot polynomials
Bigger picture
The homology theory we’ve been discussing categorifies the
polynomial corresp to Vfund = C3 .
su3
(K )
J su3 (K ) = Jfund
David Clark
Categorifying higher su3 knot polynomials
Bigger picture
The homology theory we’ve been discussing categorifies the
polynomial corresp to Vfund = C3 .
su3
(K )
J su3 (K ) = Jfund
But there are polynomials obtained by coloring a link with any
irrep Vλ of su3 .
Jλsu3 (K )
David Clark
Categorifying higher su3 knot polynomials
Bigger picture
The homology theory we’ve been discussing categorifies the
polynomial corresp to Vfund = C3 .
su3
(K )
J su3 (K ) = Jfund
But there are polynomials obtained by coloring a link with any
irrep Vλ of su3 .
Jλsu3 (K )
Ben Webster has categorified these invariants in an
algebro-geometric setting.
David Clark
Categorifying higher su3 knot polynomials
Our goal
Our goal: to categorify these higher su3 polynomials in this
local, combinatorial setting.
David Clark
Categorifying higher su3 knot polynomials
Our goal
Our goal: to categorify these higher su3 polynomials in this
local, combinatorial setting.
Possible strategies:
Categorify the su3 Jones-Wenzl idempotents.
David Clark
Categorifying higher su3 knot polynomials
Our goal
Our goal: to categorify these higher su3 polynomials in this
local, combinatorial setting.
Possible strategies:
Categorify the su3 Jones-Wenzl idempotents.
Use representation theory, and work with the symmetric
group.
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
Fix a knot K, and consider its n-parallel cable:
K
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
Fix a knot K, and consider its n-parallel cable:
K
K
David Clark
(n)
Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and
(i + 1)st cables via the right-hand rule.
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and
(i + 1)st cables via the right-hand rule.
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and
(i + 1)st cables via the right-hand rule.
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and
(i + 1)st cables via the right-hand rule.
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
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 the
Khovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n) ) −→ Kh(K (n) )
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
So let Sn act on Kh(K n ) via these maps!
David Clark
Categorifying higher su3 knot polynomials
An action of the symmetric group
So let Sn act on Kh(K n ) 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.
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Sketch of proof.
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Sketch of proof.
Our action needs to satisfy the relations on transpositions in Sn :
1
σi σj = σj σi if j 6= i ± 1
2
σi σi+1 σi = σi+1 σi σi+1
3
σi2 = 1
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Sketch of proof.
Our action needs to satisfy the relations on transpositions in Sn :
1
σi σj = σj σi if j 6= i ± 1
2
σi σi+1 σi = σi+1 σi σi+1
3
σi2 = 1
For relations (1) and (2), we need to show that
Kh(Ri Rj ) = Kh(Rj Ri )
if j 6= i ± 1
and
Kh(Ri Ri+1 Ri ) = Kh(Ri+1 Ri Ri+1 )
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K
(4)
R3 R1
K
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K
(4)
R3 R1
K
R1R3
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K
(4)
R3 R1
K
R1R3
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K
(4)
R3 R1
K
R1R3
(4)
Functoriality ⇒ Kh(Ri Rj ) = Kh(Rj Ri ).
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K
(4)
R1R2 R1
K
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K
(4)
R1R2 R1
K
R2R1R2
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K
(4)
R1R2 R1
K
R2R1R2
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K
(4)
R1R2 R1
K
R2R1R2
(4)
Functoriality ⇒ Kh(Ri Ri+1 Ri ) = Kh(Ri+1 Ri Ri+1 ).
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(Ri2 ) = Id
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(Ri2 ) = Id
K
(4)
2
R1
K
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(Ri2 ) = Id
K
(4)
2
R1
K
Id
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(Ri2 ) = Id
K
(4)
2
R1
K
Id
(4)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(Ri2 ) = Id
K
(4)
2
R1
K
Id
(4)
So we need to look more carefully at the induced maps . . .
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Mercifully, it will suffice to consider the 2-cable of K .
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Mercifully, it will suffice to consider the 2-cable of K .
In particular, we need a movie of knot diagrams that describes
the cobordism R.
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
T
x
T
T
T
x
x
x
ρ
David Clark
T
x
T
Categorifying higher su3 knot polynomials
T
Sketch of proof
T
x
T
T
T
x
x
x
ρ
T
x
T
This is a pair of R2 moves on the ends, with 4c R3 moves
in the middle (where c is the number of crossing in the
original knot K ).
David Clark
Categorifying higher su3 knot polynomials
T
Sketch of proof
T
x
T
T
T
x
x
x
ρ
T
x
T
This is a pair of R2 moves on the ends, with 4c R3 moves
in the middle (where c is the number of crossing in the
original knot K ).
That’s a very nasty map to compute explicitly!
David Clark
Categorifying higher su3 knot polynomials
T
Sketch of proof
Instead, consider the cobordism L:
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Instead, consider the cobordism L:
L
R
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Instead, consider the cobordism L:
L
R
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Instead, consider the cobordism L:
L
R
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
1
using the Categorified Kauffman Trick, and other tricks.
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
And notice that
L R
1
Id
using the Categorified Kauffman Trick, and other tricks.
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
And notice that
L R
1
Id
using the Categorified Kauffman Trick, and other tricks.
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
= Kh(L · R)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(Ri Rj ) = Kh(Rj Ri )
David Clark
X
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(Ri Rj ) = Kh(Rj Ri )
X
Kh(Ri Ri+1 Ri ) = Kh(Ri+1 Ri Ri+1 )
X
David Clark
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(Ri Rj ) = Kh(Rj Ri )
X
Kh(Ri Ri+1 Ri ) = Kh(Ri+1 Ri Ri+1 )
X
Kh(Ri2 )
David Clark
= Id
X
Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R 2 ) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(Ri Rj ) = Kh(Rj Ri )
X
Kh(Ri Ri+1 Ri ) = Kh(Ri+1 Ri Ri+1 )
X
Kh(Ri2 )
= Id
X
David Clark
Categorifying higher su3 knot polynomials
So ...
Why do we care?
David Clark
Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jλsu3 :
David Clark
Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jλsu3 :
K , Vλ
David Clark
Jλsu3 (K )
Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jλsu3 :
K , Vλ
Jλsu3 (K )
Basic idea:
1
For a knot K , we’ll find the (huge!) Khovanov complex of
one of its parallel cables.
David Clark
Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jλsu3 :
K , Vλ
Jλsu3 (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 Jλsu3 (K ).
David Clark
Categorifying higher su3 knot polynomials
Some representation theory
Let V = C3 be the standard vector representation of su3 .
David Clark
Categorifying higher su3 knot polynomials
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≥0 .
David Clark
Categorifying higher su3 knot polynomials
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≥0 .
Fact: for n = λ1 + 2λ2 , we know that Vλ is a
subrepresentation of V ⊗n .
David Clark
Categorifying higher su3 knot polynomials
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≥0 .
Fact: for n = λ1 + 2λ2 , we know that Vλ is a
subrepresentation of V ⊗n .
There is an idempotent sλ ∈ End(V ⊗n ) that projects onto
Vλ .
David Clark
Categorifying higher su3 knot polynomials
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≥0 .
Fact: for n = λ1 + 2λ2 , we know that Vλ is a
subrepresentation of V ⊗n .
There is an idempotent sλ ∈ End(V ⊗n ) that projects onto
Vλ .
The map sλ , sometimes called the Schur functor, is really
just a linear combination of permutations of the tensor
powers of V ⊗n , and can thus be viewed as
sλ ∈ QSn .
David Clark
Categorifying higher su3 knot polynomials
Some representation theory
For example, to get the adjoint representation Vad , we can
project
sad : V ⊗ V ⊗ V −→ Vad
David Clark
Categorifying higher su3 knot polynomials
Some representation theory
For example, to get the adjoint representation Vad , we can
project
sad : V ⊗ V ⊗ V −→ Vad
by letting
sad =
1
Id + τ(1 2) − τ(1 3) − τ(1 3 2)
3
David Clark
Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn , we see it acts on Kh(K (n) ).
David Clark
Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn , we see it acts on Kh(K (n) ).
V ⊗n
sλ
Vλ
David Clark
Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn , we see it acts on Kh(K (n) ).
V ⊗n
sλ
/
Vλ
David Clark
Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn , we see it acts on Kh(K (n) ).
V ⊗n
sλ
Kh(K (n) )
/
Vλ
David Clark
Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn , we see it acts on Kh(K (n) ).
V ⊗n
sλ
Kh(K (n) )
/
sλ
Vλ
“Khλ (K )”
David Clark
Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn , we see it acts on Kh(K (n) ).
V ⊗n
sλ
Kh(K (n) )
/
sλ
Vλ
“Khλ (K )”
Claim:
χ(Khλ (K )) = Jλsu3 (K )
David Clark
Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn , we see it acts on Kh(K (n) ).
V ⊗n
sλ
Kh(K (n) )
?
/
sλ
Vλ
“Khλ (K )”
Claim:
χ(Khλ (K )) = Jλsu3 (K )
David Clark
Categorifying higher su3 knot polynomials