![Page 1: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/1.jpg)
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
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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
= +
David Clark Categorifying higher su3 knot polynomials
![Page 3: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/3.jpg)
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
= +
David Clark Categorifying higher su3 knot polynomials
![Page 4: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/4.jpg)
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 polynomialcomes from coloring the link with the fundamental vectorrepresentation V ∼= C3.
David Clark Categorifying higher su3 knot polynomials
![Page 5: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/5.jpg)
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 polynomialcomes from coloring the link with the fundamental vectorrepresentation V ∼= C3.
David Clark Categorifying higher su3 knot polynomials
![Page 6: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/6.jpg)
Original categorification
Khovanov categorified this polynomial
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
![Page 7: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/7.jpg)
Original categorification
Khovanov categorified this polynomial
L1 L1Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
![Page 8: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/8.jpg)
Original categorification
Khovanov categorified this polynomial
L1 L1Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
![Page 9: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/9.jpg)
Original categorification
Khovanov categorified this polynomial
L1
L 2
L1
L2
Kh( )
Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
![Page 10: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/10.jpg)
Original categorification
Khovanov categorified this polynomial
L1
L 2
Σ
L1
L2
Kh( )
Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
![Page 11: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/11.jpg)
Original categorification
Khovanov categorified this polynomial
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
![Page 12: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/12.jpg)
“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 // •
)
David Clark Categorifying higher su3 knot polynomials
![Page 13: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/13.jpg)
“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 // •
)
David Clark Categorifying higher su3 knot polynomials
![Page 14: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/14.jpg)
“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 // •
)
David Clark Categorifying higher su3 knot polynomials
![Page 15: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/15.jpg)
Categorified spider relations (over Q)
= 0
= 3
+ + = 0
= 0
= 0
= 12 + 1
2
= 13 − 1
9 + 13 = −
David Clark Categorifying higher su3 knot polynomials
![Page 16: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/16.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 17: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/17.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 18: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/18.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 19: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/19.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 20: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/20.jpg)
Useful properties
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
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 . . .
David Clark Categorifying higher su3 knot polynomials
![Page 21: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/21.jpg)
Useful properties
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
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 . . .
David Clark Categorifying higher su3 knot polynomials
![Page 22: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/22.jpg)
Useful properties
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
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 . . .
David Clark Categorifying higher su3 knot polynomials
![Page 23: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/23.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 24: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/24.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 25: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/25.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 26: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/26.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 27: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/27.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 28: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/28.jpg)
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.
David Clark Categorifying higher su3 knot polynomials
![Page 29: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/29.jpg)
An action of the symmetric group
Fix a knot K, and consider its n-parallel cable:
K
David Clark Categorifying higher su3 knot polynomials
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An action of the symmetric group
Fix a knot K, and consider its n-parallel cable:
K K(n)
David Clark Categorifying higher su3 knot polynomials
![Page 31: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/31.jpg)
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 theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
![Page 32: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/32.jpg)
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 theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
![Page 33: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/33.jpg)
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 theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
![Page 34: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/34.jpg)
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 theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
![Page 35: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/35.jpg)
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 theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
![Page 36: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/36.jpg)
An action of the symmetric group
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.
David Clark Categorifying higher su3 knot polynomials
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An action of the symmetric group
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.
David Clark Categorifying higher su3 knot polynomials
![Page 38: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/38.jpg)
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
and
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1)
David Clark Categorifying higher su3 knot polynomials
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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
and
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1)
David Clark Categorifying higher su3 knot polynomials
![Page 40: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/40.jpg)
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
and
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1)
David Clark Categorifying higher su3 knot polynomials
![Page 41: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/41.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
![Page 42: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/42.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
![Page 43: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/43.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
![Page 44: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/44.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R R31R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
![Page 45: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/45.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R R31R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
![Page 46: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/46.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R R31R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
![Page 47: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/47.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
![Page 48: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/48.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 2R 1R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
![Page 49: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/49.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 1R2R 1R 2R 2R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
![Page 50: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/50.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 1R2R 1R 2R 2R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
![Page 51: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/51.jpg)
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 1R2R 1R 2R 2R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
![Page 52: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/52.jpg)
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 . . .
David Clark Categorifying higher su3 knot polynomials
![Page 53: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/53.jpg)
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
![Page 54: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/54.jpg)
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R Id2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
![Page 55: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/55.jpg)
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R Id2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
![Page 56: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/56.jpg)
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R Id2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
![Page 57: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/57.jpg)
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 .
David Clark Categorifying higher su3 knot polynomials
![Page 58: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/58.jpg)
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 .
David Clark Categorifying higher su3 knot polynomials
![Page 59: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/59.jpg)
Sketch of proof
T Tx Tx
Tx
ρ Tx
Tx
T
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!
David Clark Categorifying higher su3 knot polynomials
![Page 60: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/60.jpg)
Sketch of proof
T Tx Tx
Tx
ρ Tx
Tx
T
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!
David Clark Categorifying higher su3 knot polynomials
![Page 61: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/61.jpg)
Sketch of proof
T Tx Tx
Tx
ρ Tx
Tx
T
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!
David Clark Categorifying higher su3 knot polynomials
![Page 62: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/62.jpg)
Sketch of proof
Instead, consider the cobordism L:
David Clark Categorifying higher su3 knot polynomials
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Sketch of proof
Instead, consider the cobordism L:
R L
David Clark Categorifying higher su3 knot polynomials
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Sketch of proof
Instead, consider the cobordism L:
R L
David Clark Categorifying higher su3 knot polynomials
![Page 65: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/65.jpg)
Sketch of proof
Instead, consider the cobordism L:
R L
David Clark Categorifying higher su3 knot polynomials
![Page 66: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/66.jpg)
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
![Page 67: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/67.jpg)
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
And notice that
IdL R
1using the Categorified Kauffman Trick, and other tricks.David Clark Categorifying higher su3 knot polynomials
![Page 68: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/68.jpg)
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
And notice that
IdL R
1using the Categorified Kauffman Trick, and other tricks.David Clark Categorifying higher su3 knot polynomials
![Page 69: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/69.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 70: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/70.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 71: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/71.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 72: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/72.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 73: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/73.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 74: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/74.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 75: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/75.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 76: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/76.jpg)
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
![Page 77: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/77.jpg)
So ...
Why do we care?
David Clark Categorifying higher su3 knot polynomials
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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 ).
David Clark Categorifying higher su3 knot polynomials
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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 ).
David Clark Categorifying higher su3 knot polynomials
![Page 80: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/80.jpg)
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 of
one of its parallel cables.
2 Using our symmetric group action, we’ll project down to acomplex whose Euler characteristic is Jsu3λ (K ).
David Clark Categorifying higher su3 knot polynomials
![Page 81: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/81.jpg)
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 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 ).
David Clark Categorifying higher su3 knot polynomials
![Page 82: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/82.jpg)
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 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.
David Clark Categorifying higher su3 knot polynomials
![Page 83: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/83.jpg)
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 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.
David Clark Categorifying higher su3 knot polynomials
![Page 84: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/84.jpg)
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 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.
David Clark Categorifying higher su3 knot polynomials
![Page 85: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/85.jpg)
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 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.
David Clark Categorifying higher su3 knot polynomials
![Page 86: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/86.jpg)
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 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.
David Clark Categorifying higher su3 knot polynomials
![Page 87: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/87.jpg)
Some representation theory
For example, to get the adjoint representation Vad, we canproject
sad : V ⊗ V ⊗ V −→ Vad
by letting
sad =1
3
(Id + τ(1 2) − τ(1 3) − τ(1 3 2)
)
David Clark Categorifying higher su3 knot polynomials
![Page 88: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/88.jpg)
Some representation theory
For example, to get the adjoint representation Vad, we canproject
sad : V ⊗ V ⊗ V −→ Vad
by letting
sad =1
3
(Id + τ(1 2) − τ(1 3) − τ(1 3 2)
)
David Clark Categorifying higher su3 knot polynomials
![Page 89: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/89.jpg)
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
![Page 90: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/90.jpg)
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
Vλ
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
![Page 91: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/91.jpg)
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
//
Vλ
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
![Page 92: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/92.jpg)
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
Kh(K (n))
//
Vλ
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
![Page 93: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/93.jpg)
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 )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
![Page 94: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/94.jpg)
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 )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
![Page 95: Categorifying higher su3 knot polynomialsfolios.rmc.edu/davidclark/wp-content/uploads/sites/56/2016/12/uva1… · David Clark davidclark@rmc.edu Randolph-Macon College Ashland, VA](https://reader034.vdocuments.us/reader034/viewer/2022042914/5f4cf072b38a8464cd6bb6cb/html5/thumbnails/95.jpg)
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 )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials