Supercategorification and Odd Khovanov Homology Part 1 Lo - - PowerPoint PPT Presentation

Supercategorification and Odd Khovanov Homology Part 1

Léo Schelstraete 13 october 2020

1 Khovanov homology

Jones polynomial

J

• = −q−9 + q−5 + q−3 + q−1

1 Khovanov homology

Jones polynomial

J

• = −q−9 + q−5 + q−3 + q−1

Khovanov homology

Kh

• =

i

• 3
• 2
• 1

Khi Q[−9] Q[−5] Q[−3] ⊕ Q[−1]

1 Khovanov homology

Jones polynomial

J

• = −q−9 + q−5 + q−3 + q−1

Khovanov homology

Kh

• =

i

• 3
• 2
• 1

Khi Q[−9] Q[−5] Q[−3] ⊕ Q[−1]

q-graduation : qdim (Kh−3) = q−9

1 Khovanov homology

Jones polynomial

J

• = −q−9 + q−5 + q−3 + q−1

Khovanov homology

Kh

• =

i

• 3
• 2
• 1

Khi Q[−9] Q[−5] Q[−3] ⊕ Q[−1]

q-graduation : qdim (Kh−3) = q−9

(−1)0(q−3 + q−1) (−1)−2q−5 (−1)−3q−9

1 Khovanov homology

Jones polynomial

J

• = −q−9 + q−5 + q−3 + q−1

Khovanov homology

Kh

• =

i

• 3
• 2
• 1

Khi Q[−9] Q[−5] Q[−3] ⊕ Q[−1]

q-graduation : qdim (Kh−3) = q−9

(−1)0(q−3 + q−1) (−1)−2q−5 (−1)−3q−9

• i(−1)iqdim (Khi)

1 Khovanov homology

Jones polynomial

J

• = −q−9 + q−5 + q−3 + q−1

Khovanov homology

Kh

• =

i

• 3
• 2
• 1

Khi Q[−9] Q[−5] Q[−3] ⊕ Q[−1]

q-graduation : qdim (Kh−3) = q−9

(−1)0(q−3 + q−1) (−1)−2q−5 (−1)−3q−9

• i(−1)iqdim (Khi)

1 Khovanov homology

Jones polynomial

J

• = −q−9 + q−5 + q−3 + q−1

Khovanov homology

Kh

• =

i

• 3
• 2
• 1

Khi Q[−9] Q[−5] Q[−3] ⊕ Q[−1]

q-graduation : qdim (Kh−3) = q−9

(−1)0(q−3 + q−1) (−1)−2q−5 (−1)−3q−9

• i(−1)iqdim (Khi)

homology of a topological space Euler characteristic χ =

i(−1)idimHi

2 Categorification

Topological spaces

T1 T2 continuous function

f

2 Categorification

Topological spaces

T1 T2 continuous function

f H•(T1) H•(T2) H•(f ) H•

2 Categorification

Topological spaces

T1 T2 continuous function

f H•(T1) H•(T2) H•(f ) H•

K2 K1

Knots

cobordism

2 Categorification

Topological spaces

T1 T2 continuous function

f H•(T1) H•(T2) H•(f ) H•

K2 K1

Knots

cobordism

Kh(K2) Kh(K1) Kh(f ) Kh

1 + 2 Construction of Khovanov homology

Kauffman state sum of Jones polynomial: ξ0 ξ1 resolution for K: ξ ∈ {0, 1}#crossings, that is a choice of resolution ξ0 or ξ1 for each crossing. V (K) =

ξ(−1)#{ξ1 in ξ}(q + q−1)#{circles in ξ}

1 + 2 Construction of Khovanov homology

Kauffman state sum of Jones polynomial: ξ0 ξ1 resolution for K: ξ ∈ {0, 1}#crossings, that is a choice of resolution ξ0 or ξ1 for each crossing. V (K) =

ξ(−1)#{ξ1 in ξ}(q + q−1)#{circles in ξ}

1 + 2 Construction of Khovanov homology

taken from Bar-Natan, “Khovanov’s homology for tangles and cobordisms”

2′ The slice (or tangle) strategy: classical case

taken from Ohtsuki “quantum invariants”

2′ The slice (or tangle) strategy: classical case

taken from Ohtsuki “quantum invariants”

2′ The slice (or tangle) strategy: classical case

taken from Ohtsuki “quantum invariants”

2′ The slice (or tangle) strategy: classical case

taken from Ohtsuki “quantum invariants”

2′ The slice (or tangle) strategy: classical case

taken from Ohtsuki “quantum invariants”

2′ The slice (or tangle) strategy: classical case

taken from Ohtsuki “quantum invariants”

2′ The slice (or tangle) strategy: categorified case

2′ The slice (or tangle) strategy: categorified case

2′ The slice (or tangle) strategy: categorified case

· · ·

• · · ·

· · ·

• · · ·

· · ·

• · · ·

. . . . . . Ch•(S)

2′ The slice (or tangle) strategy: categorified case

· · ·

• · · ·

· · ·

• · · ·

· · ·

• · · ·

. . . . . . Ch•(S) ∗ ∗ ∗ ∗

2′ The slice (or tangle) strategy: categorified case

· · ·

• · · ·

· · ·

• · · ·

· · ·

• · · ·

. . . . . . Ch•(S) ∗ ∗ ∗ ∗ homology is an invariant ⇔ homotopy class independent of the diagram

2′ 2-categories

. . .

• .

. . . . .

• .

. . . . .

• .

. . ∗ ∗ Ch•(S)

2′ 2-categories

. . .

• .

. . . . .

• .

. . . . .

• .

. . ∗ ∗ Ch•(S) ⇒ S is a 2-category

2′ 2-categories

. . .

• .

. . . . .

• .

. . . . .

• .

. . ∗ ∗ Ch•(S) ⇒ S is a 2-category 2-categories B A

g f α

2-morphism

2′ 2-categories

. . .

• .

. . . . .

• .

. . . . .

• .

. . ∗ ∗ Ch•(S) ⇒ S is a 2-category 2-categories B A

g f α

2-morphism

1

B A

h g f β α

= B A

h f β◦α 2

C B A

g ′ f ′ β g f α

= C A

g ′g f ′f β∗α

2′ 2-categories: examples

1 homotopies: 2 natural transformations:

B A

h g f β α

= B A

h f β◦α

C B A

g ′ f ′ β g f α

= C A

g ′g f ′f β∗α

2′ Defining the invariant

− →

• ∗
• −

→

• ∗
• −

→

• ∗

− → ∗

2′ Defining the invariant

− →

• ∗
• −

→

• ∗
• −

→

• ∗

− → ∗

• 1 The complex is a cube of dimension n, where n is the number of

crossings ⇒ similar to Khovanov homology!

2′ Defining the invariant

− →

• ∗
• −

→

• ∗
• −

→

• ∗

− → ∗

• 1 The complex is a cube of dimension n, where n is the number of

crossings ⇒ similar to Khovanov homology!

2 But we used the “slice strategy”, similarly to quantum algebras,

and S is purely algebraic.

Conclusion

1 Khovanov homology is a categorication of the Jones polynomial: it categorifies the Kauffman bracket into a complex of length 1. The Jones polynomial is the Euler characteristic of this homology.

Conclusion

1 Khovanov homology is a categorication of the Jones polynomial: it categorifies the Kauffman bracket into a complex of length 1. The Jones polynomial is the Euler characteristic of this homology. 2 Categorification is the process of turning classical notion into categorical notion. We use this idea to unify the two approaches to the Jones polynomial (Khovanov homology and quantum algebras).

Conclusion

1 Khovanov homology is a categorication of the Jones polynomial: it categorifies the Kauffman bracket into a complex of length 1. The Jones polynomial is the Euler characteristic of this homology. 2 Categorification is the process of turning classical notion into categorical notion. We use this idea to unify the two approaches to the Jones polynomial (Khovanov homology and quantum algebras). 2′ The right structure to categorify the quantum algebra is a 2-category. Thanks to this structure, we can sketch a construction that match both Khovanov’s construction and the quantum algebra’s construction.

Conclusion

1 Khovanov homology is a categorication of the Jones polynomial: it categorifies the Kauffman bracket into a complex of length 1. The Jones polynomial is the Euler characteristic of this homology. 2 Categorification is the process of turning classical notion into categorical notion. We use this idea to unify the two approaches to the Jones polynomial (Khovanov homology and quantum algebras). 2′ The right structure to categorify the quantum algebra is a 2-category. Thanks to this structure, we can sketch a construction that match both Khovanov’s construction and the quantum algebra’s construction. 3 What is odd Khovanov homology? And how to adapt this construction to it (superstructures)? See you after the break!

Supercategorification and Odd Khovanov Homology Part 2

Léo Schelstraete 13 october 2020

TOP Property of odd Khovanov homology

“super” = ?

TOP Property of odd Khovanov homology

“super” = ? symmetric algebras even Khovanov homology

TOP Property of odd Khovanov homology

“super” = ? symmetric algebras even Khovanov homology exterior

• dd

TOP Property of odd Khovanov homology

“super” = ? symmetric algebras even Khovanov homology exterior

• dd

x ∧ y = (−1)|x||y|y ∧ x

TOP Property of odd Khovanov homology

“super” = ? symmetric algebras even Khovanov homology exterior

• dd

x ∧ y = (−1)|x||y|y ∧ x parity

ALG superstructures: superspaces

A superspace V is a Z/2Z-graded vector space: even and odd vectors: V = V0 ⊕ V1 |v| := grading of v (0 or 1) End(V , V ) inherits a superspace structure: even maps := maps preserving the parity

• dd maps := maps exchanging the parity

ALG superstructures: superspaces

A superspace V is a Z/2Z-graded vector space: even and odd vectors: V = V0 ⊕ V1 |v| := grading of v (0 or 1) End(V , V ) inherits a superspace structure: even maps := maps preserving the parity

• dd maps := maps exchanging the parity

super tensor product: (V ⊗W )0 = V0⊗W0⊕V1⊗W1 and (V ⊗W )1 = V0⊗W1⊕V1⊗W0 (f ⊗ g)(v ⊗ w) := (−1)|g||v|f (v) ⊗ g(w)

super interchange law (compatibility law between composition and tensor product): (f ⊗ g) ◦ (h ⊗ k) = (−1)|g||h|(f ◦ h) ⊗ (g ◦ k)

ALG superstructures: supercategories

A supercategory is a category where: each Hom-set is a superspace composition induces an even map: |f ◦ g| = |f | + |g| a superfunctor is a functor preserving parity

ALG superstructures: supercategories

A supercategory is a category where: each Hom-set is a superspace composition induces an even map: |f ◦ g| = |f | + |g| a superfunctor is a functor preserving parity A monoidal supercategory is a supercategory... ...like a monoidal category (category with a “product” like a tensor product)... ...but with the super interchange law: (f ⊗ g) ◦ (h ⊗ k) = (−1)|g||h|(f ◦ h) ⊗ (g ◦ k)

ALG superstructures: 2-supercategories

A 2-supercategory is a 2-category whose 2-morphisms have a parity... ...and compatibility between horizontal and vertical product is given by the super interchange law: C B A C B A

g ′ δ g β

• f ′

γ f α

= (−1)|β||γ| C B

g ′ f ′ δ γ

∗ B A

g f β α

(δ ∗ β) ◦ (γ ∗ α) = (−1)|β||γ|(δ ◦ γ) ∗ (β ◦ α)

TOP Product of complexes

. . .

• .

. . . . .

• .

. . . . .

• .

. . ∗ ∗ Ch•(S) S is now a 2-supercategory

TOP Product of complexes

. . .

• .

. . . . .

• .

. . . . .

• .

. . ∗ ∗ Ch•(S) S is now a 2-supercategory ⇒ Product of complexes A• ∗ B•? A• ∗ B• must preserve homotopy classes

TOP Product of complexes

. . .

• .

. . . . .

• .

. . . . .

• .

. . ∗ ∗ Ch•(S) S is now a 2-supercategory ⇒ Product of complexes A• ∗ B•? A• ∗ B• must preserve homotopy classes ⇒ Need of a definition applicable to 2-supercategories!

ALG Product of complexes

TASK: define horizontal product of complexes such that homotopy classes are preserved.

ALG Product of complexes

TASK: define horizontal product of complexes such that homotopy classes are preserved. Remark monoid ↔

• ne-object category

monoidal category ↔

• ne-object 2-category

monoidal supercategory ↔

• ne-object 2-supercategory

ALG Product of complexes

TASK: define horizontal product of complexes such that homotopy classes are preserved. Remark monoid ↔

• ne-object category

monoidal category ↔

• ne-object 2-category

monoidal supercategory ↔

• ne-object 2-supercategory

TASK2: define tensor product of complexes such that...

ALG Product of complexes

A(0) A(1)

α

⊗ B(0) B(1)

β

= A(0) ⊗ B(0) A(1) ⊗ B(0) A(0) ⊗ B(1) A(1) ⊗ B(1)

α⊗1 1⊗β 1⊗β α⊗1

ALG Product of complexes

A(0) A(1)

α

⊗ B(0) B(1)

β

= A(0) ⊗ B(0) A(1) ⊗ B(0) A(0) ⊗ B(1) A(1) ⊗ B(1)

α⊗1 1⊗β 1⊗β α⊗1

(α ⊗ 1) ◦ (1 ⊗ β) = (1 ⊗ β) ◦ (α ⊗ 1) (commutative)

ALG Product of complexes

A(0) A(1)

α

⊗ B(0) B(1)

β

= A(0) ⊗ B(0) A(1) ⊗ B(0) A(0) ⊗ B(1) A(1) ⊗ B(1)

α⊗1 1⊗β 1⊗β α⊗1

(α ⊗ 1) ◦ (1 ⊗ β) = (1 ⊗ β) ◦ (α ⊗ 1) (commutative) ⇒ Koszul rule (anti-commutative) −

ALG Product of complexes

A(0) A(1)

α

⊗ B(0) B(1)

β

= A(0) ⊗ B(0) A(1) ⊗ B(0) A(0) ⊗ B(1) A(1) ⊗ B(1)

α⊗1 1⊗β 1⊗β α⊗1

(α ⊗ 1) ◦ (1 ⊗ β) = (1 ⊗ β) ◦ (α ⊗ 1) (commutative) ⇒ Koszul rule (anti-commutative) − Assume α and β are odd: (α ⊗ 1) ◦ (1 ⊗ β) = −(1 ⊗ β) ◦ (α ⊗ 1) (anti-commutative)

ALG Product of complexes

A(0) A(1)

α

⊗ B(0) B(1)

β

= A(0) ⊗ B(0) A(1) ⊗ B(0) A(0) ⊗ B(1) A(1) ⊗ B(1)

α⊗1 1⊗β 1⊗β α⊗1

(α ⊗ 1) ◦ (1 ⊗ β) = (1 ⊗ β) ◦ (α ⊗ 1) (commutative) ⇒ Koszul rule (anti-commutative) − Assume α and β are odd: (α ⊗ 1) ◦ (1 ⊗ β) = −(1 ⊗ β) ◦ (α ⊗ 1) (anti-commutative) ⇒ super Koszul rule?

ALG Product of complexes

Theorem The super Koszul rule:

1 exists for homogeneous complexes (complexes whose differentials

are either even or odd)

2 is unique, at least for cubes (all choices of signs result in isomorphic

complexes)

3 preserves homotopy classes: given complexes A•, B•, C• and D•, if

A• ≃ B• and C• ≃ D• Then A• ⊗ C• ≃ B• ⊗ D•

TOP Definition of the invariant

− → − → − → − →

TOP Definition of the invariant

− → − → − → − →

• F(1, 1)
• E(0, 2)
• (1, 1)

even q−1

EF(1, 1)

• q

EF(1, 1)

• dd

(1, 1)

Conclusion

1 We defined a knot invariant using a supercategorification S of a

quantum algebra...

Conclusion

1 We defined a knot invariant using a supercategorification S of a

quantum algebra...

2 Is it odd Khovanov homology? Still a conjecture but...

coincide on simple examples (and differ from even) as odd Khovanov homology, the invariant can be split into two identical invariants: Khodd = Kh′

• dd ⊕ Kh′
• dd.

Conclusion

1 We defined a knot invariant using a supercategorification S of a

quantum algebra...

2 Is it odd Khovanov homology? Still a conjecture but...

coincide on simple examples (and differ from even) as odd Khovanov homology, the invariant can be split into two identical invariants: Khodd = Kh′

• dd ⊕ Kh′
• dd.

3 Application? Proof of functoriality