Tangle sums and factorization of A -polynomials Masaharu ISHIKAWA Tohoku University RIMS Seminar in Hakone, 1 June 2012 Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 1 / 31
Plan of this talk § 1. Factorization of A -polynomials § 2. Alexander polynomials and epimorphisms § 3. Cyclic surgeries Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 2 / 31
§ 1. Factorization of A -polynomials K : a knot in S 3 M K : the complement of K ı ∗ : X ( M K ) → X ( ∂M K ) : induced by the ı # : π 1 ( ∂M K ) → π 1 ( M K ) Λ ⊂ R ( ∂M K ) : the set of diagonal representations of π 1 ( ∂M K ) t | Λ : Λ → X ( ∂M K ) p : Λ → C ∗ × C ∗ : taking the left-top entries of ρ ( µ ) and ρ ( λ ) X 1 , · · · , X k : irreducible components of X ( M K ) p · t | − 1 alg. closure in X ( ∂M K ) ı ∗ → ı ∗ ( X i ) Λ X i − —————————— − → Y i —— − → D i A i ( L, M ) : the defining equation of D i Definition The A -polynomial of a knot K is defined as k ∏ A K ( L, M ) = A i ( L, M ) . i =1 Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 3 / 31
Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 4 / 31
Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 5 / 31
Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 6 / 31
Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 7 / 31
Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 8 / 31
Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 9 / 31
Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 10 / 31
9 24 is given by the sum of tangles 1 / 3 + ( − 1 / 3) and 5 / 2 . T S 9 37 is given by the sum of tangles 1 / 3 + ( − 1 / 3) and 5 / 3 . Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 11 / 31
Theorem (Mattman-Shimokawa-I., 2011) Suppose that N ( T ) and N ( S + T ) are knots and N ( S ) is a split link in S 3 . Then A ◦ N ( T ) ( L, M ) | A N ( S + T ) ( L, M ) Here A ◦ K ( L, M ) is the product of factors of A K ( L, M ) containing the variable L . x x x x x x z z S T T y y y y y y w w Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 12 / 31
§ 2. Alexander polynomials and epimorphisms RT R A K fac. type Alex. poly. epi. (3 1 ) 3 8 10 1 / 3 , 3 / 2 , − 1 / 3 3 1 A → 3 1 8 11 [2 , − 2 , 3 , 2 , − 2] 3 1 B (3 1 )(6 1 ) No (3 1 ) 2 (4 1 ) 9 24 1 / 3 , 5 / 2 , − 1 / 3 4 1 A → 3 1 9 37 1 / 3 , 5 / 3 , − 1 / 3 4 1 B (4 1 )(6 1 ) → 4 1 10 21 [2 , − 2 , 5 , 2 , − 2] 5 1 B (5 1 )(6 1 ) No 10 40 [2 , 2 , 3 , − 2 , − 2] 3 1 B (3 1 )(8 8 ) → 3 1 (continued) T T S = S Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 13 / 31
Table: Factorizations of RT R knots (2nd page) RT R A K fac. type Alex. poly. epi. (3 1 )(4 1 ) 2 10 59 2 / 5 , 3 / 2 , − 2 / 5 3 1 A → 4 1 (3 1 ) 2 (5 1 ) 10 62 1 / 3 , 5 / 4 , − 1 / 3 5 1 A → 3 1 (3 1 ) 2 (5 2 ) 10 65 1 / 3 , 7 / 4 , − 1 / 3 5 2 A → 3 1 10 67 1 / 3 , 7 / 5 , − 1 / 3 5 2 B (5 2 )(6 1 ) No 10 74 1 / 3 , 7 / 3 , − 1 / 3 5 2 B (5 2 )(6 1 ) → 5 2 (3 1 ) 2 (5 2 ) 10 77 1 / 3 , 7 / 2 , − 1 / 3 5 2 A → 3 1 (3 1 ) 2 (6 1 ) 10 98 1 / 3 , T 0 , − 1 / 3 3 1 #3 1 B → 3 1 3 1 #3 mir (3 1 ) 4 10 99 1 / 3 , T 1 , − 1 / 3 A → 3 1 1 (3 1 ) 3 10 143 1 / 3 , 3 / 4 , − 1 / 3 3 1 A → 3 1 10 147 1 / 3 , 3 / 5 , − 1 / 3 3 1 B (3 1 )(6 1 ) No Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 14 / 31
Lemma. Let K = N ( R + T + R ) be an RT R knot with R = R ( p/q ) and q > 0 . Then (i) q > 1 . (ii) If K is of type A then ∆ K ( t ) = ∆ N ( T ) ( t )∆ D ( R ) ( t ) 2 . (iii) If K is of type B then ∆ K ( t ) = ∆ N ( T ) ( t )∆ N ( R+R(1/1)+ ) ( t ) . R (iv) The knot determinant of K is divisible by q 2 . Proposition Let K be a prime knot of 10 or fewer crossings. Suppose that K is not 8 18 , 9 40 , 10 82 , 10 87 , or 10 103 . Then K is RT R with N ( T ) a non-trivial knot of 10 or fewer crossings if and only if it is in the above table. Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 15 / 31
Definition An epimorphism φ : π 1 ( M K 1 ) → π 1 ( M K 2 ) is said to be preserving peripheral structures if φ ( π 1 ( ∂M K 1 )) ⊂ π 1 ( ∂M K 2 ) . Theorem (Hoste-Shanahan, 2010) Suppose that there exists an epimorphism φ : π 1 ( M K 1 ) → π 1 ( M K 2 ) preserving peripheral structures. Then φ ( µ 1 ) = µ 2 and φ ( λ 1 ) = λ d 2 for some d ∈ Z . A K 2 ( L, M ) | ( L d − 1) A K 1 ( L d , M ) . Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 16 / 31
RT R A K fac. type Alex. poly. epi. (3 1 ) 2 (4 1 ) 9 24 1 / 3 , 5 / 2 , − 1 / 3 4 1 A → 3 1 y y x x y x x x x T x x x x y x y y x Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 17 / 31
RT R A K fac. type Alex. poly. epi. 8 11 [2 , − 2 , 3 , 2 , − 2] 3 1 B (3 1 )(6 1 ) No 9 37 1 / 3 , 5 / 3 , − 1 / 3 4 1 B (4 1 )(6 1 ) → 4 1 y y x y x x T x x y y y x Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 18 / 31
RT R A K fac. type Alex. poly. epi. 9 37 1 / 3 , 5 / 3 , − 1 / 3 4 1 B (4 1 )(6 1 ) → 4 1 Fact (Kitano-Suzuki, 2008) There exists an epimorphism φ : π 1 ( M 9 37 ) → π 1 ( M 4 1 ) such that 81¯ 8¯ 2 , 72¯ 8¯ 3 , 94¯ 9¯ 3 , 34¯ 3¯ 5 , 15¯ 1¯ 5 , 56¯ 5¯ 7 , 27¯ 2¯ 8 , 49¯ 4¯ π 1 ( M 9 37 ) 8 (1 , ¯ 8¯ 79¯ 31¯ 5¯ ( µ 1 , λ 1 ) 2461) (2 , ¯ 12¯ ( µ 2 , λ 2 ) 34) 1 �→ 2 , 2 �→ 3 , 3 �→ 14¯ φ 1 , 4 �→ 3 , 5 �→ 1 , 6 �→ ¯ 141 , 7 �→ 4 , 8 �→ 1 , 9 �→ 4 ¯ 43¯ φ ( λ ) 21 = − λ By Hoste-Shanahan, A 4 1 ( L, M ) | A 9 37 ( L, M ) . Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 19 / 31
K 2 ,k : (2 , k ) -torus knot. Corollary (Mattman-Shimokawa-I., 2011) Let K be the 2 -bridge knot described below, where k > 2 is odd and n > 1 . Then π 1 ( M K ) admits no epimorphism onto π 1 ( M K 2 ,k ) preserving peripheral structure, although A K 2 ,k ( L, M ) | A K ( L, M ) . k crossings n crossings n crossings T First assertion follows from Gonz´ alez-Acu˜ na - Ram´ ırez. Second assertion is a corollary of our factorization. Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 20 / 31
§ 3. Cyclic surgeries M : a compact, connected, irreducible and ∂ -irreducible 3 -manifold whose boundary ∂M is a torus. R ( M ) = Hom ( π 1 ( M ) , SL (2 , C )) X ( M ) : the character variety of M R ( M ) ∋ ρ �→ χ ρ ∈ X ( M ) : the character of ρ γ ∈ π 1 ( M ) I γ : X ( M ) → C : the regular function defined by I γ ( χ ρ ) = χ ρ ( γ ) f γ : X ( M ) → C : defined by f γ = I 2 γ − 4 . Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 21 / 31
Definition A 1 -dimensional algebraic subset X 1 of X ( M ) is called a norm curve if f α is not constant for any α ∈ H 1 ( ∂M, Z ) \ { 0 } . X 1 : a norm curve ˜ X 1 : the smooth model of the projective completion of X 1 α ∈ π 1 ( ∂M ) � α � X 1 : the degree of f α on ˜ X 1 Lemma � · � X 1 is a norm. Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 22 / 31
X ( i ) : irreducible component of X 1 1 d ( i ) : X ( i ) : the degree of the map ı ∗ | X ( i ) → X ( ∂M ) 1 1 1 Definition The A -polynomial of X 1 with multiplicity is defined as k 1 ( L, M ) d ( i ) A ( i ) A d ∏ 1 . 1 ( L, M ) = i =1 Theorem (Boyer-Zhang, 2001) � · � X 1 = � · � A d 1 . Example: A 4 1 ( L, M ) = M 4 + L ( − 1 + M 2 + 2 M 4 + M 6 − M 8 ) + L 2 M 4 Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 23 / 31
The next results follow immediately from CGLS. Theorem Suppose that N ( S + T ) is a knot, N ( T ) is a hyperbolic knot and N ( S ) is a split link in S 3 . Let X 0 be the irreducible component of X ( M N ( T ) ) containing the character of a discrete faithful representation of π 1 ( M N ( T ) ) . If α is not a strict boundary class of N ( T ) associated with an ideal point of X 0 and satisfies � α � X 0 > � µ � X 0 then π 1 ( M N ( S + T ) ( α )) is not cyclic as well as π 1 ( M N ( T ) ( α )) is not. Corollary Suppose further that N ( S + T ) is a small knot. If every α ∈ H 1 ( ∂M N ( T ) ; Z ) \ { 0 } except strict boundary classes of N ( T ) satisfies � α � X 0 > � µ � X 0 then N ( S + T ) has no non-trivial cyclic slope. Masaharu ISHIKAWA (Tohoku University) Factorization of A -polynomials Talk in Hakone 24 / 31
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