INVARIANTS OF VIRTUAL LINKS Julia Mikhalchishina NOVOSIBIRSK 2015 - PowerPoint PPT Presentation

INVARIANTS OF VIRTUAL LINKS Julia Mikhalchishina NOVOSIBIRSK – 2015

Braid group B n = � σ 1 , . . . , σ n − 1 � – the braid group . σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for i = 1 , 2 , . . . , n − 2 , = for | i − j | ≥ 2 . σ i σ j σ j σ i

Braid group B n = � σ 1 , . . . , σ n − 1 � – the braid group . σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for i = 1 , 2 , . . . , n − 2 , = for | i − j | ≥ 2 . σ i σ j σ j σ i σ i σ − 1 = 1 i

Braid group B n = � σ 1 , . . . , σ n − 1 � – the braid group . σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for i = 1 , 2 , . . . , n − 2 , = for | i − j | ≥ 2 . σ i σ j σ j σ i σ i σ − 1 = 1 i ( σ 2 σ − 1 1 ) n Example:

Knots and links A knot S 1 − → S 3

Knots and links A knot S 1 − → S 3 An n -component link S 1 × . . . × S 1 → S 3 − � �� � n

The connection between braids and knots Alexander’s theorem. Given a link L then ∃ β ∈ B n : L = � β.

The connection between braids and knots Alexander’s theorem. Given a link L then ∃ β ∈ B n : L = � β. Рис.: The trefoil T = � σ 3 1

Markov’s theorem. Given braids β 1 , β 2 ∈ B n then M 1 , M 2 β 1 = � � β 2 ⇐ ⇒ β 1 − − − − → β 2 β ↔ σ i βσ − 1 M 1 i = 1 , 2 , . . . , n − 2 , (1) i β ↔ βσ ± 1 β ∈ B n , βσ ± 1 M 2 ∈ B n + 1 . (2) n n

Group of the link G ( L ) = π 1 ( S 3 \ N ( L )) .

Group of the link G ( L ) = π 1 ( S 3 \ N ( L )) . "Braid method". Given L = � β G ( L ) = � x 1 , . . . , x n � ϕ A ( β )( x i ) = x i , i = 1 , 2 , . . . , n � , where ϕ A : B n − → Aut ( F n ) – the Artin representation  x i �→ x i x i + 1 x − 1 i ,  ϕ A ( σ i ) : x i + 1 �→ x i ,  x j �→ x j , j � = i , i + 1 .

Virtual braid group VB n = � B n , S n � = � σ 1 , . . . , σ n − 1 , ρ 1 , . . . , ρ n − 1 � – the virtual braid group . � σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for i = 1 , 2 , . . . , n − 2 , I = for | i − j | ≥ 2 . σ i σ j σ j σ i

Virtual braid group VB n = � B n , S n � = � σ 1 , . . . , σ n − 1 , ρ 1 , . . . , ρ n − 1 � – the virtual braid group . � σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for i = 1 , 2 , . . . , n − 2 , I = for | i − j | ≥ 2 . σ i σ j σ j σ i  ρ i ρ i + 1 ρ i = ρ i + 1 ρ i ρ i + 1 for i = 1 , 2 , . . . , n − 2 ,  for | i − j | ≥ 2 , II ρ i ρ j = ρ j ρ i  ρ 2 = 1 for i = 1 , 2 , ..., n − 2 . i

Virtual braid group VB n = � B n , S n � = � σ 1 , . . . , σ n − 1 , ρ 1 , . . . , ρ n − 1 � – the virtual braid group . � σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for i = 1 , 2 , . . . , n − 2 , I = for | i − j | ≥ 2 . σ i σ j σ j σ i  ρ i ρ i + 1 ρ i = ρ i + 1 ρ i ρ i + 1 for i = 1 , 2 , . . . , n − 2 ,  for | i − j | ≥ 2 , II ρ i ρ j = ρ j ρ i  ρ 2 = 1 for i = 1 , 2 , ..., n − 2 . i � ρ i ρ i + 1 σ i = for i = 1 , 2 , . . . , n − 2 , σ i + 1 ρ i ρ i + 1 III σ i ρ j = ρ j σ i for | i − j | ≥ 2 .

Virtual braid group VB n = � B n , S n � = � σ 1 , . . . , σ n − 1 , ρ 1 , . . . , ρ n − 1 � – the virtual braid group . � σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for i = 1 , 2 , . . . , n − 2 , I = for | i − j | ≥ 2 . σ i σ j σ j σ i  ρ i ρ i + 1 ρ i = ρ i + 1 ρ i ρ i + 1 for i = 1 , 2 , . . . , n − 2 ,  for | i − j | ≥ 2 , II ρ i ρ j = ρ j ρ i  ρ 2 = 1 for i = 1 , 2 , ..., n − 2 . i � ρ i ρ i + 1 σ i = for i = 1 , 2 , . . . , n − 2 , σ i + 1 ρ i ρ i + 1 III σ i ρ j = ρ j σ i for | i − j | ≥ 2 .

Bardakov, Bellingeri; Manturov ψ : VB n − → Aut ( F n + 1 ) , F n + 1 = � x 1 , . . . , x n , y � .

Bardakov, Bellingeri; Manturov ψ : VB n − → Aut ( F n + 1 ) , F n + 1 = � x 1 , . . . , x n , y � . ψ | B n = ϕ A .   x i �→ x i x i + 1 x − 1 x i �→ yx i + 1 y − 1 , i ,       x i + 1 �→ y − 1 x i y , x i + 1 �→ x i , ψ ( σ i ) : ψ ( ρ i ) :  x j �→ x j , j � = i , i + 1 ,  x j �→ x j , j � = i , i + 1 ,     y �→ y . y �→ y .

Group of the virtual link Let L v = � β v , β v ∈ VB n , G ( L v ) = � x 1 , . . . , x n , y � ψ ( β v )( x i ) = x i , i = 1 , 2 , . . . , n � .

Group of the virtual link Let L v = � β v , β v ∈ VB n , G ( L v ) = � x 1 , . . . , x n , y � ψ ( β v )( x i ) = x i , i = 1 , 2 , . . . , n � . Example: The virtual trefoil T v = � σ 2 1 ρ 1

Group of the virtual link Let L v = � β v , β v ∈ VB n , G ( L v ) = � x 1 , . . . , x n , y � ψ ( β v )( x i ) = x i , i = 1 , 2 , . . . , n � . Example: The virtual trefoil T v = � σ 2 1 ρ 1 G ( T v ) = G ( � 1 ρ 1 ) = � x 1 , x 2 , y � ψ ( σ 2 1 ρ 1 )( x 1 ) = x 1 , ψ ( σ 2 σ 2 1 ρ 1 )( x 2 ) = x 2 � .

Wada representations w r 1 , w 2 , w 3 : B n − → Aut ( F n ) .  x i → x r i x i + 1 x − r ,  i w r 1 ( σ i ) : x i + 1 → x i ,  x j → x j , for j � = i , i + 1 , r ∈ Z , r � = 0 . Note for r = 1 this is the Artin representation.  x i → x i x − 1 i + 1 x i ,  w 2 ( σ i ) : x i + 1 → x i ,  x j → x j , for j � = i , i + 1 .  x i → x 2 i x i + 1 ,  x i + 1 → x − 1 i + 1 x − 1 w 3 ( σ i ) : x i + 1 , i  x j → x j , for j � = i , i + 1 .

Wada representations w 1 , w 2 , w 3 : B n − → Aut ( F n ) .

Wada representations w 1 , w 2 , w 3 : B n − → Aut ( F n ) . We construct the mappings W k : VB n − → Aut ( F n + 1 ) , k = 1 , 2 , 3 , W k | B n = w k .  x i �→ yx i + 1 y − 1 ,  x i + 1 �→ y − 1 x i y , W k ( ρ i ) :  x j �→ x j , for j � = i , i + 1 .

Wada representations w 1 , w 2 , w 3 : B n − → Aut ( F n ) . We construct the mappings W k : VB n − → Aut ( F n + 1 ) , k = 1 , 2 , 3 , W k | B n = w k .  x i �→ yx i + 1 y − 1 ,  x i + 1 �→ y − 1 x i y , W k ( ρ i ) :  x j �→ x j , for j � = i , i + 1 . Proposition. Constructed mappings W k , k = 1 , 2 , 3 , are representations of VB n − → AutF n + 1 .

Let L v = � β v , β v ∈ VB n , k = 1 , 2 , 3 , G k ( L v ) = � x 1 , . . . , x n , y � W k ( β v )( x i ) = x i , i = 1 , 2 , . . . , n � .

Let L v = � β v , β v ∈ VB n , k = 1 , 2 , 3 , G k ( L v ) = � x 1 , . . . , x n , y � W k ( β v )( x i ) = x i , i = 1 , 2 , . . . , n � . Theorem. Groups G k ( L v ) are invariants of the virtual link L v , k = 1 , 2 , 3.

Markov Theorem for virtuals Theorem (Kauffman, Lambropoulou). Given braids β 1 , β 2 ∈ VB n then K 1 , K 2 , K 3 , K 4 β 1 = � � β 2 ⇐ ⇒ β 1 − − − − − − − − → β 2

Markov Theorem for virtuals Theorem (Kauffman, Lambropoulou). Given braids β 1 , β 2 ∈ VB n then K 1 , K 2 , K 3 , K 4 β 1 = � � β 2 ⇐ ⇒ β 1 − − − − − − − − → β 2 ρ k β v ρ k ∼ β v ∼ σ k β v σ − 1 K1) Virtual and real conjugation: k ,

Markov Theorem for virtuals Theorem (Kauffman, Lambropoulou). Given braids β 1 , β 2 ∈ VB n then K 1 , K 2 , K 3 , K 4 β 1 = � � β 2 ⇐ ⇒ β 1 − − − − − − − − → β 2 ρ k β v ρ k ∼ β v ∼ σ k β v σ − 1 K1) Virtual and real conjugation: k , β v ρ n ∼ β v ∼ β v σ ± 1 K2) Right virtual and real stabilization: n ,

Markov Theorem for virtuals Theorem (Kauffman, Lambropoulou). Given braids β 1 , β 2 ∈ VB n then K 1 , K 2 , K 3 , K 4 β 1 = � � β 2 ⇐ ⇒ β 1 − − − − − − − − → β 2 ρ k β v ρ k ∼ β v ∼ σ k β v σ − 1 K1) Virtual and real conjugation: k , β v ρ n ∼ β v ∼ β v σ ± 1 K2) Right virtual and real stabilization: n , β v ∼ β v σ ± 1 n ρ n − 1 σ ∓ 1 K3) Algebraic right over/under threading: n ,

Markov Theorem for virtuals Theorem (Kauffman, Lambropoulou). Given braids β 1 , β 2 ∈ VB n then K 1 , K 2 , K 3 , K 4 β 1 = � � β 2 ⇐ ⇒ β 1 − − − − − − − − → β 2 ρ k β v ρ k ∼ β v ∼ σ k β v σ − 1 K1) Virtual and real conjugation: k , β v ρ n ∼ β v ∼ β v σ ± 1 K2) Right virtual and real stabilization: n , β v ∼ β v σ ± 1 n ρ n − 1 σ ∓ 1 K3) Algebraic right over/under threading: n , K4) Algebraic left over/under threading: β v ∼ β v ρ n ρ n − 1 σ ∓ 1 n − 1 ρ n σ ± 1 n − 1 ρ n − 1 ρ n , где β v , ρ k , σ k ∈ VB n , k = 1 , . . . , n − 1, а ρ n , σ n ∈ VB n + 1 .

w 1 , w 2 , w 3 : B n − → Aut ( F n ) .

w 1 , w 2 , w 3 : B n − → Aut ( F n ) . � W 1 , W 2 , W 3 : VB n − → Aut ( F n + 1 ) [ Proposition. ]

w 1 , w 2 , w 3 : B n − → Aut ( F n ) . � W 1 , W 2 , W 3 : VB n − → Aut ( F n + 1 ) [ Proposition. ] � Let L v = � β v , β v ∈ VB n , k = 1 , 2 , 3 , G k ( L v ) = � x 1 , . . . , x n , y � W k ( β v )( x i ) = x i , i = 1 , 2 , . . . , n � .