Epimorphisms between knot groups: determination of the partial order - PowerPoint PPT Presentation

Epimorphisms between knot groups: determination of the partial order Masaaki Suzuki Akita University September 14, 2010 Macky Epimorphisms between knot groups

Twisted Alexander polynomial G : a finitely presentable group α : G ։ Z l : a surjective homomorphism of G ρ : G − → GL ( n ; R ) : a representation of G R : UFD ∆ N G,ρ = ⇒ ∆ G,ρ = : twisted Alexander polynomial ∆ D G,ρ Macky Epimorphisms between knot groups

� � Theorem. (Kitano-S.-Wada) G, G ′ : finitely presentable groups α : G ։ Z l , α ′ : G ′ ։ Z l : surjective homomorphisms s.t. α = α ′ ◦ φ , ∃ φ : G ։ G ′ ⇒ ∆ N G,ρ can be divided by ∆ N G ′ ,ρ ′ and ∆ D G,ρ = ∆ D = G ′ ,ρ ′ for any representation ρ ′ : G ′ − → GL ( n ; R ) , where ρ = ρ ′ ◦ φ . ϕ ϕ � � G ′ � � G ′ G G � � � � ��������� � � � � � � � � � � � � � � � � α ρ � � α ′ � ρ ′ � � � � � � � GL ( n ; R ) Z l Macky Epimorphisms between knot groups

Example α : G ։ Z l , G = ⟨ x 1 , . . . , x u | r 1 , . . . , r v ⟩ , ρ : G → GL ( n, R ) Macky Epimorphisms between knot groups

Example α : G ։ Z l , G = ⟨ x 1 , . . . , x u | r 1 , . . . , r v ⟩ , ρ : G → GL ( n, R ) G ′ = ⟨ x 1 , . . . , x u | r 1 , . . . , r v , s ⟩ π : G ։ G ′ : the projection Macky Epimorphisms between knot groups

� � Example α : G ։ Z l , G = ⟨ x 1 , . . . , x u | r 1 , . . . , r v ⟩ , ρ : G → GL ( n, R ) G ′ = ⟨ x 1 , . . . , x u | r 1 , . . . , r v , s ⟩ π : G ։ G ′ : the projection Suppose that s ∈ ker α, s ∈ ker ρ ϕ ϕ G ′ G ′ G � � G � � � � � � ��������� � � � � � � � � � � � � � � � � α ρ � � ∃ α ′ � ∃ ρ ′ � � � � � � � Z l GL ( n ; R ) ⇒ ∆ N G,ρ can be divided by ∆ N G ′ ,ρ ′ and ∆ D G,ρ = ∆ D = G ′ ,ρ ′ Macky Epimorphisms between knot groups

Twisted Alexander polynomial G : a finitely presentable group α : G ։ Z l : a surjective homomorphism ρ : G − → GL ( n ; R ) : a representation of G R : UFD = ⇒ ∆ G,ρ : twisted Alexander polynomial Macky Epimorphisms between knot groups

Twisted Alexander polynomial G : a finitely presentable group α : G ։ Z l : a surjective homomorphism ρ : G − → GL ( n ; R ) : a representation of G R : UFD = ⇒ ∆ G,ρ : twisted Alexander polynomial Twisted Alexander polynomial for knots G ( K ) : the knot group of a knot K α : G ( K ) ։ Z : the abelianization → SL (2; Z /p Z ) : a representation ρ : G ( K ) − p : prime = ⇒ ∆ K,ρ : twisted Alexander polynomial for the knot K Macky Epimorphisms between knot groups

K : a knot, ρ : G ( K ) − → SL (2; Z /p Z ) ∆ K,ρ : the twisted Alexander polynomial of K ∆ N K,ρ : the numerator of ∆ K,ρ ∆ D K,ρ : the denominator of ∆ K,ρ Corollary. If there exists a representation ρ ′ : G ( K ′ ) → SL (2; Z /p Z ) such that for any representation ρ : G ( K ) → SL (2; Z /p Z ) , ∆ N K,ρ can not be divided by ∆ N K ′ ,ρ ′ or ∆ D K,ρ ̸ = ∆ D K ′ ,ρ ′ , ⇒ there exists no epimorphism G ( K ) ։ G ( K ′ ) . = Macky Epimorphisms between knot groups

Corollary. If there exists a representation ρ ′ : G ( K ′ ) → SL (2; Z /p Z ) such that for any representation ρ : G ( K ) → SL (2; Z /p Z ) , ∆ N K,ρ can not be divided by ∆ N K ′ ,ρ ′ or ∆ D K,ρ ̸ = ∆ D K ′ ,ρ ′ , ⇒ there exists no epimorphism G ( K ) ։ G ( K ′ ) . = K : a knot, ∆ K : the Alexander polynomial of K Fact. K, K ′ : two knots If ∆ K can not be divided by ∆ K ′ , ⇒ there exists no epimorphism G ( K ) ։ G ( K ′ ) . = Macky Epimorphisms between knot groups

K : a prime knot in S 3 i.e. G ( K ) = π 1 ( S 3 − K ) G ( K ) : the knot group of K Definition. K ≥ K ′ ⇐ ∃ φ : G ( K ) ։ G ( K ′ ) ⇒ Macky Epimorphisms between knot groups

K : a prime knot in S 3 i.e. G ( K ) = π 1 ( S 3 − K ) G ( K ) : the knot group of K Definition. K ≥ K ′ ⇐ ∃ φ : G ( K ) ։ G ( K ′ ) ⇒ Fact. The relation “ ≥ ” is a partial order on the set of prime knots. • K ≥ K • K ≥ K ′ , K ′ ≥ K = ⇒ K = K ′ • K ≥ K ′ , K ′ ≥ K ′′ = ⇒ K ≥ K ′′ Macky Epimorphisms between knot groups

Theorem (Horie-Kitano-Matsumoto-S.) The partial order “ ≥ ” on the set of prime knots with up to 11 crossings is given by  8 5 , 8 10 , 8 15 , 8 18 , 8 19 , 8 20 , 8 21 , 9 1 , 9 6 , 9 16 , 9 23 , 9 24 , 9 28 , 9 40 ,   10 5 , 10 9 , 10 32 , 10 40 , 10 61 , 10 62 , 10 63 , 10 64 , 10 65 , 10 66 , 10 76 ,     10 77 , 10 78 , 10 82 , 10 84 , 10 85 , 10 87 , 10 98 , 10 99 , 10 103 , 10 106 ,     10 112 , 10 114 , 10 139 , 10 140 , 10 141 , 10 142 , 10 143 , 10 144 ,      10 159 , 10 164 ,     11 a 43 , 11 a 44 , 11 a 46 , 11 a 47 , 11 a 57 , 11 a 58 , 11 a 71 , 11 a 72 , 11 a 73 ,     11 a 100 , 11 a 106 , 11 a 107 , 11 a 108 , 11 a 109 , 11 a 117 , 11 a 134 , 11 a 139 ,  ≥ 3 1 11 a 157 , 11 a 165 , 11 a 171 , 11 a 175 , 11 a 176 , 11 a 194 , 11 a 196 , 11 a 203 ,   11 a 212 , 11 a 216 , 11 a 223 , 11 a 231 , 11 a 232 , 11 a 236 , 11 a 244 , 11 a 245 ,     11 a 261 , 11 a 263 , 11 a 264 , 11 a 286 , 11 a 305 , 11 a 306 , 11 a 318 , 11 a 332 ,     11 a 338 , 11 a 340 , 11 a 351 , 11 a 352 , 11 a 355 , 11 n 71 , 11 n 72 , 11 n 73 ,     11 n 74 , 11 n 75 , 11 n 76 , 11 n 77 , 11 n 78 , 11 n 81 , 11 n 85 , 11 n 86 , 11 n 87 ,      11 n 94 , 11 n 104 , 11 n 105 , 11 n 106 , 11 n 107 , 11 n 136 , 11 n 164 , 11 n 183 ,     11 n 184 , 11 n 185  Macky Epimorphisms between knot groups

 8 18 , 9 37 , 9 40 ,   10 58 , 10 59 , 10 60 , 10 122 , 10 136 , 10 137 , 10 138 ,  ≥ 4 1 11 a 5 , 11 a 6 , 11 a 51 , 11 a 132 , 11 a 239 , 11 a 297 , 11 a 348 , 11 a 349 ,   11 n 100 , 11 n 148 , 11 n 157 , 11 n 165  11 n 78 , 11 n 148 ≥ 5 1 10 74 , 10 120 , 10 122 , 11 n 71 , 11 n 185 ≥ 5 2 11 a 352 ≥ 6 1 11 a 351 ≥ 6 2 11 a 47 , 11 a 239 ≥ 6 3 Macky Epimorphisms between knot groups

To determine the partial order on the set of prime knots For each pair of two prime knots K, K ′ , determine whether there exists an epimorphism φ : G ( K ) ։ G ( K ′ ) Macky Epimorphisms between knot groups

To determine the partial order on the set of prime knots For each pair of two prime knots K, K ′ , determine whether there exists an epimorphism φ : G ( K ) ։ G ( K ′ ) The number of prime knots with up to 11 crossings is 801 . Then the number of cases to consider is 801 P 2 = 640 , 800 . Macky Epimorphisms between knot groups

To determine the partial order on the set of prime knots For each pair of two prime knots K, K ′ , determine whether there exists an epimorphism φ : G ( K ) ։ G ( K ′ ) The number of prime knots with up to 11 crossings is 801 . Then the number of cases to consider is 801 P 2 = 640 , 800 . The number of prime knots with up to 12 crossings is 2 , 977 . Then the number of cases to consider is 2977 P 2 = 4 , 429 , 776 . Macky Epimorphisms between knot groups

To prove the existence of an epimorphism Constructing an epimorphism explicitly Macky Epimorphisms between knot groups

To prove the existence of an epimorphism Constructing an epimorphism explicitly i.e. ? ∃ φ : G (8 18 ) ։ G (3 1 ) 8 18 ≥ 3 1 ? Example. 8 18 = , 3 1 = ⟨ x 1 , x 2 , x 3 , � x 4 x 1 ¯ x 4 ¯ x 2 , x 5 x 3 ¯ x 5 ¯ x 2 , x 6 x 3 ¯ x 6 ¯ x 4 , ⟩ � x 4 , x 5 , x 6 , x 7 x 5 ¯ x 7 ¯ x 4 , x 8 x 5 ¯ x 8 ¯ x 6 , x 1 x 7 ¯ x 1 ¯ x 6 , G (8 18 ) = � � x 7 , x 8 x 2 x 7 ¯ x 2 ¯ x 8 � G (3 1 ) = ⟨ y 1 , y 2 , y 3 | y 3 y 1 ¯ y 3 ¯ y 2 , y 1 y 2 ¯ y 1 ¯ y 3 ⟩ Macky Epimorphisms between knot groups

To prove the existence of an epimorphism Constructing an epimorphism explicitly i.e. ? ∃ φ : G (8 18 ) ։ G (3 1 ) 8 18 ≥ 3 1 ? Example. 8 18 = , 3 1 = ⟨ x 1 , x 2 , x 3 , � x 4 x 1 ¯ x 4 ¯ x 2 , x 5 x 3 ¯ x 5 ¯ x 2 , x 6 x 3 ¯ x 6 ¯ x 4 , ⟩ � x 4 , x 5 , x 6 , x 7 x 5 ¯ x 7 ¯ x 4 , x 8 x 5 ¯ x 8 ¯ x 6 , x 1 x 7 ¯ x 1 ¯ x 6 , G (8 18 ) = � � x 7 , x 8 x 2 x 7 ¯ x 2 ¯ x 8 � G (3 1 ) = ⟨ y 1 , y 2 , y 3 | y 3 y 1 ¯ y 3 ¯ y 2 , y 1 y 2 ¯ y 1 ¯ y 3 ⟩ φ ( x 1 ) = y 1 , φ ( x 2 ) = y 2 , φ ( x 3 ) = y 1 , φ ( x 4 ) = y 3 , φ ( x 5 ) = y 3 , φ ( x 6 ) = y 1 y 3 ¯ y 1 , φ ( x 7 ) = y 3 , φ ( x 8 ) = y 1 8 18 ≥ 3 1 Macky Epimorphisms between knot groups

To prove the non-existence of any epimorphism (1) By the (classical) Alexander polynomial K : a knot ∆ K : the Alexander polynomial of K Fact. K, K ′ : two knots If ∆ K can not be divided by ∆ K ′ , ⇒ there exists no epimorphism G ( K ) ։ G ( K ′ ) . = Macky Epimorphisms between knot groups

i.e. ? ∃ φ : G (4 1 ) ։ G (8 21 ) Example. 4 1 ≥ 8 21 ? 4 1 = , 8 21 = ∆ 4 1 = t 2 − 3 t + 1 , ∆ 8 21 = t 4 − 4 t 3 + 5 t 2 − 4 t + 1 Macky Epimorphisms between knot groups

i.e. ? ∃ φ : G (4 1 ) ։ G (8 21 ) Example. 4 1 ≥ 8 21 ? 4 1 = , 8 21 = ∆ 4 1 = t 2 − 3 t + 1 , ∆ 8 21 = t 4 − 4 t 3 + 5 t 2 − 4 t + 1 t 2 − 3 t + 1 ∆ 4 1 = t 4 − 4 t 3 + 5 t 2 − 4 t + 1 ∆ 8 21 ∆ 8 21 can not divide ∆ 4 1 4 1 � 8 21 Macky Epimorphisms between knot groups