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 � � = − q − 9 + q − 5 + q − 3 + q − 1 J polynomial

1 Khovanov homology Khovanov � � i -3 -2 -1 0 Kh = homology Kh i Q [ − 9 ] Q [ − 5 ] 0 Q [ − 3 ] ⊕ Q [ − 1 ] Jones � � = − q − 9 + q − 5 + q − 3 + q − 1 J polynomial

1 Khovanov homology Khovanov � � i -3 -2 -1 0 Kh = homology Kh i Q [ − 9 ] Q [ − 5 ] 0 Q [ − 3 ] ⊕ Q [ − 1 ] q -graduation : qdim ( Kh − 3 ) = q − 9 Jones � � = − q − 9 + q − 5 + q − 3 + q − 1 J polynomial

1 Khovanov homology Khovanov � � i -3 -2 -1 0 Kh = homology Kh i Q [ − 9 ] Q [ − 5 ] 0 Q [ − 3 ] ⊕ Q [ − 1 ] q -graduation : qdim ( Kh − 3 ) = q − 9 ( − 1 ) 0 ( q − 3 + q − 1 ) ( − 1 ) − 2 q − 5 ( − 1 ) − 3 q − 9 Jones � � = − q − 9 + q − 5 + q − 3 + q − 1 J polynomial

1 Khovanov homology Khovanov � � i -3 -2 -1 0 Kh = homology Kh i Q [ − 9 ] Q [ − 5 ] 0 Q [ − 3 ] ⊕ Q [ − 1 ] q -graduation : qdim ( Kh − 3 ) = q − 9 i ( − 1 ) i qdim ( Kh i ) � ( − 1 ) 0 ( q − 3 + q − 1 ) ( − 1 ) − 2 q − 5 ( − 1 ) − 3 q − 9 Jones � � = − q − 9 + q − 5 + q − 3 + q − 1 J polynomial

1 Khovanov homology Khovanov � � i -3 -2 -1 0 Kh = homology Kh i Q [ − 9 ] Q [ − 5 ] 0 Q [ − 3 ] ⊕ Q [ − 1 ] q -graduation : qdim ( Kh − 3 ) = q − 9 i ( − 1 ) i qdim ( Kh i ) � ( − 1 ) 0 ( q − 3 + q − 1 ) ( − 1 ) − 2 q − 5 ( − 1 ) − 3 q − 9 Jones � � = − q − 9 + q − 5 + q − 3 + q − 1 J polynomial

1 Khovanov homology Khovanov � � homology of a i -3 -2 -1 0 Kh = homology topological space Kh i Q [ − 9 ] Q [ − 5 ] 0 Q [ − 3 ] ⊕ Q [ − 1 ] q -graduation : qdim ( Kh − 3 ) = q − 9 i ( − 1 ) i qdim ( Kh i ) i ( − 1 ) i dim H i � χ = � ( − 1 ) 0 ( q − 3 + q − 1 ) ( − 1 ) − 2 q − 5 ( − 1 ) − 3 q − 9 Jones � � Euler = − q − 9 + q − 5 + q − 3 + q − 1 J polynomial characteristic

2 Categorification Topological spaces T 1 T 2 f continuous function

2 Categorification Topological spaces T 1 T 2 f continuous function H • H • ( f ) H • ( T 1 ) H • ( T 2 )

2 Categorification Knots Topological spaces K 1 K 2 T 1 T 2 f cobordism continuous function H • H • ( f ) H • ( T 1 ) H • ( T 2 )

2 Categorification Knots Topological spaces K 1 K 2 T 1 T 2 f cobordism continuous function H • Kh Kh ( f ) H • ( f ) Kh ( K 1 ) Kh ( K 2 ) H • ( T 1 ) H • ( T 2 )

1 + 2 Construction of Khovanov homology Kauffman state sum of Jones polynomial: ξ 0 ξ 1 resolution for K : ξ ∈ { 0 , 1 } # crossings , that is a choice of resolution ξ 0 or ξ 1 for each crossing. ξ ( − 1 ) # { ξ 1 in ξ } ( q + q − 1 ) # { circles in ξ } V ( K ) = �

1 + 2 Construction of Khovanov homology Kauffman state sum of Jones polynomial: ξ 0 ξ 1 resolution for K : ξ ∈ { 0 , 1 } # crossings , that is a choice of resolution ξ 0 or ξ 1 for each crossing. ξ ( − 1 ) # { ξ 1 in ξ } ( q + q − 1 ) # { circles in ξ } V ( K ) = �

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 homotopy class ⇔ is an invariant independent of the diagram

2 ′ 2-categories Ch • ( S ) . . . . . . . . . • • • ∗ ∗ • • • . . . . . . . . .

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

2 ′ 2-categories g Ch • ( S ) 2-categories . . . B A α . . . . . . f 2-morphism • • • ∗ ∗ • • • . . . . . . . . . ⇒ S is a 2-category

2 ′ 2-categories g Ch • ( S ) 2-categories . . . B A α . . . . . . f 2-morphism • • • h ∗ ∗ h β • • • g B A = B A 1 β ◦ α . . . α . . . f . . . f g ′ g ′ g g ⇒ S is a 2-category = C 2 C β B α A β ∗ α A f ′ f f ′ f

2 ′ 2-categories: examples 1 homotopies: 2 natural transformations: h g ′ g ′ g g h β g B A = B β ◦ α A C B α A = C β ∗ α A β α f f ′ f f ′ f f

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 exterior algebras even odd Khovanov homology

TOP Property of odd Khovanov homology “super” = ? symmetric exterior algebras even odd Khovanov x ∧ y = ( − 1 ) | x || y | y ∧ x homology

TOP Property of odd Khovanov homology “super” = ? parity symmetric exterior algebras even odd Khovanov x ∧ y = ( − 1 ) | x || y | y ∧ x homology