Kauffman brackets on surfaces Francis Bonahon University of - PowerPoint PPT Presentation

Kauffman brackets on surfaces Kauffman brackets on surfaces Francis Bonahon University of Southern California Geometric Topology in New York, August 2013 1/28

Kauffman brackets on surfaces Joint work with Helen Wong 2/28

Kauffman brackets on surfaces Joint work with Helen Wong here with Grace Tsapsie Hibbard, born March 22, 2013 2/28

Kauffman brackets on surfaces SL 2 ( C )–characters S = closed oriented surface of genus g � 0 group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) 3/28

Kauffman brackets on surfaces SL 2 ( C )–characters S = closed oriented surface of genus g � 0 group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) 3/28

Kauffman brackets on surfaces SL 2 ( C )–characters S = closed oriented surface of genus g � 0 A group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) defines its character K ρ : { closed curves in S } − → C K �− → Tr ρ ( K ) 3/28

Kauffman brackets on surfaces SL 2 ( C )–characters S = closed oriented surface of genus g � 0 A group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) defines its character K ρ : { closed multicurves in S } − → C n n � → ( − 1) n � K = K i �− Tr ρ ( K i ) i =1 i =1 3/28

Kauffman brackets on surfaces SL 2 ( C )–characters Theorem (Helling 1967) A function K : { closed multicurves in S } − → C is the character of a group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) if and only if: 4/28

Kauffman brackets on surfaces SL 2 ( C )–characters Theorem (Helling 1967) A function K : { closed multicurves in S } − → C is the character of a group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) if and only if: ◮ (Homotopy Invariance) K ( K ) depends only on the homotopy class of K 4/28

Kauffman brackets on surfaces SL 2 ( C )–characters Theorem (Helling 1967) A function K : { closed multicurves in S } − → C is the character of a group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) if and only if: ◮ (Homotopy Invariance) K ( K ) depends only on the homotopy class of K ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) K ( K 2 ) 4/28

Kauffman brackets on surfaces SL 2 ( C )–characters Theorem (Helling 1967) A function K : { closed multicurves in S } − → C is the character of a group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) if and only if: ◮ (Homotopy Invariance) K ( K ) depends only on the homotopy class of K ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) K ( K 2 ) ◮ (Skein Relation) K ( K 1 ) = −K ( K 0 ) − K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 4/28

Kauffman brackets on surfaces SL 2 ( C )–characters Theorem (Helling 1967) A function K : { closed multicurves in S } − → C is the character of a group homomorphism ρ : π 1 ( S ) → SL 2 ( C ) if and only if: ◮ (Homotopy Invariance) K ( K ) depends only on the homotopy class of K ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) K ( K 2 ) ◮ (Skein Relation) K ( K 1 ) = −K ( K 0 ) − K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = The Skein Relation just rephrases the classical trace relation of SL 2 ( C ): Tr M Tr N = Tr MN + Tr MN − 1 , ∀ M , N ∈ SL 2 ( C ) 4/28

Kauffman brackets on surfaces SL 2 ( C )–characters Definition An SL 2 ( C ) –character is a function K : { closed multicurves in S } − → C such that: ◮ (Homotopy Invariance) K ( K ) depends only on the homotopy class of K ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) K ( K 2 ) for any multicurves K 1 and K 2 ◮ (Skein Relation) K ( K 1 ) = −K ( K 0 ) − K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 5/28

Kauffman brackets on surfaces Kauffman brackets Definition For q = e 2 π i � ∈ C − { 0 } , a Kauffman q–bracket is a function K : { framed links in S × [0 , 1] } − → End ( E ) for a finite-dimensional vector space E, such that: ◮ (Isotopy Invariance) K ( K ) depends only on the isotopy class of K in S × [0 , 1] ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) ◦ K ( K 2 ) whenever K = K 1 ∪ K 2 with K 1 ⊂ S × [0 , 1 2 ] and K 2 ⊂ S × [ 1 2 , 1] 1 2 K ( K 0 ) + q − 1 ◮ (Skein Relation) K ( K 1 ) = q 2 K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 6/28

Kauffman brackets on surfaces Kauffman brackets Definition For q = e 2 π i � ∈ C − { 0 } , a Kauffman q–bracket is a function K : { framed links in S × [0 , 1] } − → End ( E ) for a finite-dimensional vector space E, such that: ◮ (Isotopy Invariance) K ( K ) depends only on the isotopy class of K in S × [0 , 1] ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) ◦ K ( K 2 ) whenever K = K 1 ∪ K 2 with K 1 ⊂ S × [0 , 1 2 ] and K 2 ⊂ S × [ 1 2 , 1] 1 2 K ( K 0 ) + q − 1 ◮ (Skein Relation) K ( K 1 ) = q 2 K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 6/28

Kauffman brackets on surfaces Kauffman brackets Definition For q = e 2 π i � ∈ C − { 0 } , a Kauffman q–bracket is a function K : { framed links in S × [0 , 1] } − → End ( E ) for a finite-dimensional vector space E, such that: ◮ (Isotopy Invariance) K ( K ) depends only on the isotopy class of K in S × [0 , 1] ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) ◦ K ( K 2 ) whenever K = K 1 ∪ K 2 with K 1 ⊂ S × [0 , 1 2 ] and K 2 ⊂ S × [ 1 2 , 1] 1 2 K ( K 0 ) + q − 1 ◮ (Skein Relation) K ( K 1 ) = q 2 K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 6/28

Kauffman brackets on surfaces Kauffman brackets Definition For q = e 2 π i � ∈ C − { 0 } , a Kauffman q–bracket is a function K : { framed links in S × [0 , 1] } − → End ( E ) for a finite-dimensional vector space E, such that: ◮ (Isotopy Invariance) K ( K ) depends only on the isotopy class of K in S × [0 , 1] ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) ◦ K ( K 2 ) whenever K = K 1 ∪ K 2 with K 1 ⊂ S × [0 , 1 2 ] and K 2 ⊂ S × [ 1 2 , 1] 1 2 K ( K 0 ) + q − 1 ◮ (Skein Relation) K ( K 1 ) = q 2 K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 6/28

Kauffman brackets on surfaces Kauffman brackets Definition For q = e 2 π i � ∈ C − { 0 } , a Kauffman q–bracket is a function → End ( E ) ∼ K : { framed links in S × [0 , 1] } − = M n ( C ) for a finite-dimensional vector space E, such that: ◮ (Isotopy Invariance) K ( K ) depends only on the isotopy class of K in S × [0 , 1] ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) ◦ K ( K 2 ) whenever K = K 1 ∪ K 2 with K 1 ⊂ S × [0 , 1 2 ] and K 2 ⊂ S × [ 1 2 , 1] 1 2 K ( K 0 ) + q − 1 ◮ (Skein Relation) K ( K 1 ) = q 2 K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 6/28

Kauffman brackets on surfaces Kauffman brackets Definition For q = e 2 π i � ∈ C − { 0 } , a Kauffman q–bracket is a function → End ( E ) ∼ K : { framed links in S × [0 , 1] } − = M n ( C ) for a finite-dimensional vector space E, such that: ◮ (Isotopy Invariance) K ( K ) depends only on the isotopy class of K in S × [0 , 1] ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) ◦ K ( K 2 ) whenever K = K 1 ∪ K 2 with K 1 ⊂ S × [0 , 1 2 ] and K 2 ⊂ S × [ 1 2 , 1] 1 2 K ( K 0 ) + q − 1 ◮ (Skein Relation) K ( K 1 ) = q 2 K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 6/28

Kauffman brackets on surfaces Kauffman brackets Definition For q = e 2 π i � ∈ C − { 0 } , a Kauffman q–bracket is a function → End ( E ) ∼ K : { framed links in S × [0 , 1] } − = M n ( C ) for a finite-dimensional vector space E, such that: ◮ (Isotopy Invariance) K ( K ) depends only on the isotopy class of K in S × [0 , 1] ◮ (Superposition Rule) K ( K 1 ∪ K 2 ) = K ( K 1 ) ◦ K ( K 2 ) whenever K = K 1 ∪ K 2 with K 1 ⊂ S × [0 , 1 2 ] and K 2 ⊂ S × [ 1 2 , 1] 1 2 K ( K 0 ) + q − 1 ◮ (Skein Relation) K ( K 1 ) = q 2 K ( K ∞ ) if K 1 , K 0 , K ∞ are the same everywhere, except in a small box where K 1 = , K 0 = and K ∞ = 6/28

Kauffman brackets on surfaces Kauffman brackets Historic examples 1. When S = the sphere and End ( E ) = End ( C ) = C , the only example is the classical Kauffman bracket ( ∼ = Jones polynomial) K : { framed links in R 3 } − → C 7/28

Kauffman brackets on surfaces Kauffman brackets Historic examples 1. When S = the sphere and End ( E ) = End ( C ) = C , the only example is the classical Kauffman bracket ( ∼ = Jones polynomial) K : { framed links in R 3 } − → C 2. Witten’s interpretation (1987) of the Jones polynomial in the framework of a topological quantum field theory, mathematicalized by Reshetikhin-Turaev, provides a Kauffman q –bracket K WRT : { framed links in S × [0 , 1] } − → End ( E ) for every q that is an N –root of unity with N odd. 7/28