On the twisted Alexander polynomial for metabelian SL 2 ( C ) - PowerPoint PPT Presentation

On the twisted Alexander polynomial for metabelian SL 2 ( C ) –representations with the adjoint action Yoshikazu Yamaguchi JSPS Research fellow (PD) Tokyo Institute of Technology RIMS Seminar “Twisted topological invariants and topology of low-dimensional manifolds” Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Basic notion and notation The twisted Alexander polynomial The twisted A refinement of ∆ K ( t ) Alexander = (the Alexander polynomial) with ρ : π 1 → GL ( V ) polynomial Notation E K := S 3 \ N ( K ) a knot exterior, ρ Ad SL 2 ( C ) Ad ◦ ρ : π 1 ( E K ) − → − − → Aut ( sl 2 ( C )) Ad ρ ( γ ) : v �→ ρ ( γ ) v ρ ( γ ) − 1 γ �→ ρ ( γ ) �→ � 0 1 � 1 0 � 0 0 � � � sl 2 ( C ) = C ⊕ C ⊕ C 0 − 1 0 0 1 0 The adjoint action Ad gives a connection with the character variety Hom ( π 1 ( E K ) , SL 2 ( C )) // SL 2 ( C ) . Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Metabelian representations Definition of metabelian reps. ρ : π 1 ( E K ) → SL 2 ( C ) is metabelian ρ ([ π 1 ( E K ) , π 1 ( E K )]) ⊂ SL 2 ( C ) an abelian subgroup. ⇐ ⇒ Remark ρ : π 1 ( E K ) → SL 2 ( C ) is abelian ρ ( π 1 ( E K )) ⊂ SL 2 ( C ) ⇐ ⇒ an abelian subgroup, ⇐ ⇒ ρ ([ π 1 ( E K ) , π 1 ( E K )]) = { 1 } , ρ ⇐ ⇒ SL 2 ( C ) π 1 ( E K ) H 1 ( E K ; Z ) π 1 ( E K ) / [ π 1 ( E K ) , π 1 ( E K )] ≃ � µ � ρ is determined by only ρ ( µ ) . We focus on non–abelian representations. Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Definitions of reducible and irreducible reps. ρ : π 1 ( E K ) → SL 2 ( C ) ρ : reducible Def ⇒ ∃ L ⊂ C 2 ⇐ s.t. ρ ( g )( L ) ⊂ L ( ∀ g ∈ π 1 ( E K )) by taking conjugation, �� �� a � ∗ ⊂ SL 2 ( C ) � ρ : π 1 ( E K ) → � a ∈ C \ { 0 } a − 1 � 0 ρ : irreducible Def ⇐ ⇒ ρ : not reducible. Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Reducible and irreducible reps. in metabelian ones ρ : π 1 ( E K ) → SL 2 ( C ) metabelian ρ : reducible �� � � 1 � ω ⊂ SL 2 ( C ) � ρ : [ π 1 ( E K ) , π 1 ( E K )] → ± � ω ∈ C � 0 1 �� a �� � ∗ since Im ρ ⊂ � � a ∈ C \ { 0 } and a − 1 � 0 � 1 � ∗ A , B ∈ SL 2 ( C ) : upper triangular ⇒ ABA − 1 B − 1 = 0 1 ρ : irreducible �� �� a � 0 ⊂ SL 2 ( C ) � ρ : [ π 1 ( E K ) , π 1 ( E K )] → � a ∈ C \ { 0 } a − 1 � 0 � 0 � 1 ρ ( µ ) = by F. Nagasato. − 1 0 Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Background ρ : π 1 ( E K ) → SL 2 ( C ) metabelian ρ : reducible ∆ K ( t ) appears in the twisted Alexander polynomial. → Hyperbolic torsion at “bifurcation points” − in Hom ( π 1 ( E K , SL 2 ( C ))) // SL 2 ( C ) . ρ : irreducible “Does ∆ K ( t ) appear in the twisted Alexander?” Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main Theorem Theorem Suppose that � an irred. metabelian and; ρ : π 1 ( E K ) → SL 2 ( C ) s.t. “ longitude–regular ” . (the twisted Alexander poly.) · ∆ α ⊗ Ad ◦ ρ Then ( t ) = ( t − 1 )∆ K ( − t ) P ( t ) , E K where ∆ K ( t ) is the Alexander polynomial of K. Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Hierarchy of metabelian representations Remark ρ : π 1 ( E K ) → SL 2 ( C ) reducible = ⇒ ρ : metabelian �� a � 1 ∗ ( ∵ ) Im ρ ⊂ ∗ � a � = 0 � � � � �� ⇒ ρ ([ π 1 ( E K ) , π 1 ( E K )]) ⊂ ± 0 a − 1 0 1 We have the following hierarchy of metabelian representations: abelian ⊂ reducible ⊂ metabelian difference difference roots of ∆ K ( t ) ∆ K ( − 1 ) Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The details of reducible metabelian representations Existence of reducible representations (G. Burde, G. de Rham) ∃ ρ : π 1 ( E K ) → SL 2 ( C ) : reducible and non–abelian if and only if ∆ K ( λ 2 ) = 0 where λ is an eigenvalue of ρ ( µ ) . { the characters of non–abelian reducible representations } Hom ( π 1 ( E K ) , SL 2 ( C )) // SL 2 ( C ) , = C abel ∩ C non–abel in where C abel is the component of abelian characters and C non–abel is the components of non–abelian ones. Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The twisted Alexander polynomial of reducible SL 2 ( C ) -representations Theorem ρ : π 1 ( E K ) → SL 2 ( C ) reducible and non–abelian, � λ is an eigenvalue of ρ ( µ ) and corresponding to λ ∈ C s.t. , ∆ K ( λ 2 ) = 0 then ( t ) = ∆ K ( λ 2 t ) ( t − 1 ) · ∆ K ( λ − 2 t ) ( λ 2 t − 1 ) · ∆ K ( t ) ∆ α ⊗ Ad ◦ ρ ( λ − 2 t − 1 ) . E K Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The details of irreducible metabelian representations Existence of irreducible metabelian reps. (Nagasato) ♯ { the characters of irereducible metabelian reps. } = | ∆ K ( − 1 ) | − 1 . 2 ι ∈ H 1 ( E K ; Z 2 ) induces ι on Hom ( π 1 ( E K ) , SL 2 ( C )) // SL 2 ( C ) . an involution ˆ Remark (Nagasato & Y.) { the characters of irereducible metabelian reps. } in Hom ( π 1 ( E K ) , SL 2 ( C )) // SL 2 ( C ) . = { the fixed point set of ˆ ι } Remark A higher rank analogy was given by H. Boden and S. Friedl. Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main Theorem (again) Theorem Suppose that � an irred. metabelian and; ρ : π 1 ( E K ) → SL 2 ( C ) s.t. “ longitude–regular ” . (the twisted Alexander poly.) · ∆ α ⊗ Ad ◦ ρ Then ( t ) = ( t − 1 )∆ K ( − t ) P ( t ) , E K where ∆ K ( t ) is the Alexander polynomial of K. Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The details of ∆ α ⊗ Ad ◦ ρ ( t ) E K Homomorphisms ρ : π 1 ( E K ) → SL 2 ( C ) is metabelian ρ ([ π 1 ( E K ) , π 1 ( E K )])( ⊂ SL 2 ( C )) is abelian, ⇐ ⇒ Suppose that ρ ([ π 1 ( E K ) , π 1 ( E K )]) � = { 1 } . α : π 1 ( E K ) → π 1 ( E K ) / [ π 1 ( E K ) , π 1 ( E K )] ≃ H 1 ( E K ) = � t � s.t. α ( µ ) = t the twisted Alexander poly. � � � � ∂ r i det α ⊗ Ad ◦ ρ ∂ g j i , j � = 1 ∆ α ⊗ Ad ◦ ρ ( t ) = E K det ( α ⊗ Ad ◦ ρ ( g 1 − 1 )) from a presentation π 1 ( E K ) = � g 1 , g 2 , . . . , g k | r 1 , . . . , r k − 1 � . Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools We need a “good” presentation of π 1 ( E K ) for metabelian reps. X-S. Lin introduced a suitable presentation of π 1 ( E K ) by using a free Seifert surface of K . a Seifert surface S is free ⇔ S 3 = S × [ − 1 , 1 ] ∪ S 3 \ S × [ − 1 , 1 ] : a Heegaard splitting. Figure: a free Seifert surface of the trefoil knot Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools We need a “good” presentation of π 1 ( E K ) for metabelian reps. X-S. Lin introduced a suitable presentation of π 1 ( E K ) by using a free Seifert surface of K . a Seifert surface S is free ⇔ S 3 = S × [ − 1 , 1 ] ∪ S 3 \ S × [ − 1 , 1 ] : a Heegaard splitting. Figure: a free Seifert surface of the trefoil knot Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools We need a “good” presentation of π 1 ( E K ) for metabelian reps. X-S. Lin introduced a suitable presentation of π 1 ( E K ) by using a free Seifert surface of K . a Seifert surface S is free ⇔ S 3 = S × [ − 1 , 1 ] ∪ S 3 \ S × [ − 1 , 1 ] : a Heegaard splitting. Figure: a free Seifert surface of the trefoil knot Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools We need a “good” presentation of π 1 ( E K ) for metabelian reps. X-S. Lin introduced a suitable presentation of π 1 ( E K ) by using a free Seifert surface of K . a Seifert surface S is free ⇔ S 3 = S × [ − 1 , 1 ] ∪ S 3 \ S × [ − 1 , 1 ] : a Heegaard splitting. Figure: a free Seifert surface of the trefoil knot Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Lin’s presentation By using a free Seifert surface S with genus 2 g , i µ − 1 = a − π 1 ( E K ) = � µ, x 1 , . . . , x 2 g | µ a + i ( i = 1 , . . . 2 g ) � where x i is a closed loop corresponding to 1–handle in S 3 \ S × [ − 1 , 1 ] , a ± i is a word in x 1 , . . . , x 2 g , corresponding to closed loops in the spine of S . Figure: a free Seifert surface of the trefoil knot Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Lin’s presentation By using a free Seifert surface S with genus 2 g , i µ − 1 = a − π 1 ( E K ) = � µ, x 1 , . . . , x 2 g | µ a + i ( i = 1 , . . . 2 g ) � where x i is a closed loop corresponding to 1–handle in S 3 \ S × [ − 1 , 1 ] , a ± i is a word in x 1 , . . . , x 2 g , corresponding to closed loops in the spine of S . Figure: a free Seifert surface of the trefoil knot Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.