VIEWS: 19 PAGES: 19 POSTED ON: 9/19/2012
Normalization of Twisted Alexander Invariants Takahiro Kitayama∗ Abstract Twisted Alexander invariants of knots are well-deﬁned up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and deﬁne normal- ized twisted Alexander invariants. We can show that the invariants coincide with sign- determined Reidemeister torsion in a normalized setting and reﬁne the duality theorem. As an application, we obtain stronger necessary conditions for a knot to be ﬁbered than those previously known. Finally, we study a behavior of the highest degree of the normalized invariant. 2000 Mathematics Subject Classiﬁcation. Primary: 57M25, Secondary: 57M05; 57Q10 Keywords and phrases. twisted Alexander invariant, Reidemeister torsion, ﬁbered knot 1 Introduction Twisted Alexander invariants, which coincide with Reidemeister torsion ([Ki], [KL]), were introduced for knots in the 3-sphere by Lin [L] and generally for ﬁnitely presentable groups by Wada [Wad]. They were given a natural topological deﬁnition by using twisted homology groups in the notable work of Kirk and Livingston [KL]. Many properties of the classical Alexander polynomial ∆K were subsequently extended to the twisted case and it was shown that the invariants have much information on the topological structure of a space. For example, necessary conditions of twisted Alexander invariants for a knot to be ﬁbered were given by Cha [C], Goda-Morifuji [GM], Goda-Kitano-Morifuji [GKM] and Friedl-Kim [FK]. Moreover, even suﬃcient conditions for a knot with genus 1 to be ﬁbered were obtained by Friedl-Vidussi [FV]. It is well known that ∆K can be normalized, for instance, by considering the skein relation. In this paper, we ﬁrst obtain the corresponding result in twisted settings. The twisted Alexander invariant ∆K,ρ associated to a linear representation ρ is well-deﬁned up to multiplication of units in a Laurent polynomial ring. We show that the ambiguity can be eliminated via a combinatorial method constructed by Wada and deﬁne the normalized twisted Alexander invariant ∆K,ρ (See Deﬁnition 4.4 and Theorem 4.5). ∗ The author is supported by Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists. 1 2 T. Kitayama Turaev [T2] deﬁned sign-determined Reidemeister torsion by reﬁning the sign ambiguity of Reidemeister torsion for a odd-dimensional manifold and showed that the other ambiguity depends on the choice of Euler structures. We also normalize sign-determined Reidemeister torsion T K,ρ for a knot and deﬁne T K,ρ (t). Then we prove the equality ∆K,ρ (t) = T K,ρ (t) (Theorem 5.7). This shows that ∆K,ρ is a simple homotopy invariant and give rise to a reﬁned version (Theorem 5.9) of the duality theorem for twisted Alexander invariants. As an application, we generalize above results for ﬁbered knots. We can deﬁne the highest degree and the coeﬃcient of the highest degree term of ∆K,ρ . We show that these values are completely determined for ﬁbered knots (See Theorem 6.3). Finally, we obtain the following inequality which bounds free genus g f (K) from below by using the highest degree h-deg ∆K,ρ : 2 h-deg ∆K,ρ ≤ n(2g f (K) − 1). (1.1) (See Theorem 6.6.) This paper is organized as follows. In the next section, we ﬁrst review the deﬁnition of twisted Alexander invariants for knots. We also describe how to compute them from a presen- tation of a knot group and the duality theorem for unitary representations. In Section 3, we review Turaev’s sign-determined Reidemeister torsion and the relation with twisted Alexander invariants. In Section 4, we establish normalization of twisted Alexander invariants. In Section 5, we reﬁne the correspondence with sign-determined Reidemeister torsion and the duality the- orem for twisted Alexander invariants. Section 6 is devoted to applications. Here we generalize the result of Cha [C], Goda-Kitano-Morifuji [GKM] and Friedl-Kim [FK] for ﬁbered knots and study a behavior of the highest degree and obtain (1.1). Acknowledgement. The author would like to express his gratitude to Toshitake Kohno for his encouragement and helpful suggestions. He also would like to thank Hiroshi Goda, Teru- aki Kitano, Takayuki Morifuji and Yoshikazu Yamaguchi for fruitful discussions and advice and Stefan Friedl for several stimulating comments which lead to some improvements of the argument in this revised version. 2 Twisted Alexander invariants In this section, we review twisted Alexander invariants of K following [C] and [KL]. For a given oriented knot K in S 3 , let E K := S 3 \ N(K), where N(K) denotes an open tubular neighborhood of K and G K := π1 E K . We ﬁx an element µ ∈ G K represented by a meridian of E K and denote by α : G K → t be the abelianization homomorphism which maps µ to the generator t. Let R be a Noetherian unique factorization domain and Q(R) the quotient ﬁeld of R. We ﬁrst give a deﬁnition of a twisted homology group and a twisted cohomology group. Let X be a connected CW-complex and X the universal covering of X. The chain complex C∗ (X) is Normalization of twisted Alexander invariants 3 a left Z[π1 X]-module via the action of π1 X as the deck transformations of X. We regard C∗ (X) also as a right Z[π1 X]-module by deﬁning σ · γ := γ−1 · σ, where γ ∈ π1 X and σ ∈ C∗ (X). For a linear representation ρ : π1 X → GLn (R), R⊕n naturally has a left Z[π1 X]-module structure. We deﬁne the twisted homology group Hi (X; R⊕n ) and the twisted cohomology group H i (X; R⊕n ) of ρ ρ ρ as follows: Hi (X; R⊕n ) := Hi (C∗ (X) ⊗Z[π1 X] R⊕n ), ρ H i (X; R⊕n ) := H i (HomZ[π1 X] (C∗ (X), R⊕n )). ρ Deﬁnition 2.1. For a representation ρ : G K → GLn (R), we deﬁne ∆iK,ρ to be the order of the i-th twisted homology group Hi (E K ; R[t, t−1 ]⊕n ), where we consider R[t, t−1 ]⊕n = R[t, t−1 ] ⊗ R⊕n . α⊗ρ It is called the i-th twisted Alexander polynomial associated to ρ, which is well-deﬁned up to multiplication by a unit in R[t, t−1 ]. We furthermore deﬁne ∆K,ρ := ∆1 /∆0 ∈ Q(R)(t). K,ρ K,ρ It is called the twisted Alexander invariant associated to ρ and well-deﬁned up to a factor ηtl , where η ∈ R× and l ∈ Z. Remark 2.2. Lin’s twisted Alexander polynomial deﬁned in [L] coincides with ∆1 . K,ρ The homomorphisms α and α ⊗ ρ induce ring homomorphisms α : Z[G K ] → Z[t, t−1 ] and ˜ −1 Φ : Z[G K ] → Mn (R[t, t ]). For a knot diagram of K, we choose and ﬁx a Wirtinger presentation G K = x1 , . . . , xm | r1 , . . . , rm−1 . Let us consider the (m − 1) × m matrix AΦ whose component ∂r ∂ is the n × n matrix Φ ∂xij ∈ Mn (R[t, t−1 ]), where ∂x j denotes Fox’s free derivative with respect to x j . For 1 ≤ k ≤ m, let us denote by AΦ,k the (m − 1) × (m − 1) matrix obtained from AΦ by removing the k-th column. We regard AΦ,k as an (m − 1)n × (m − 1)n matrix with coeﬃcients in R[t, t−1 ]. The twisted Alexander invariants can be computed from a Wirtinger presentation as follows. This is nothing but Wada’s construction in [Wad]. Theorem 2.3 ([HLN], [KL]). For a representation ρ : G K → GLn (R) and a Wirtinger presenta- tion x1 , . . . , xm | r1 , . . . , rm−1 of G K , we have det AΦ,k ∆K,ρ ≡ mod ηtl η∈R× ,l∈Z det Φ(xk − 1) for any index k. Remark 2.4. Wada shows in [Wad] that the twisted Alexander invariant is well-deﬁned up to a factor ηtln . He also shows that in case that ρ is a unimodular representation, the twisted Alexander invariant is well-deﬁned up to a factor ±tln if n is odd and up to only tln if n is even. It is also known that the twisted Alexander invariants have the following duality. We extend complex conjugation to C(t) by taking t → t−1 . Theorem 2.5 ([Ki], [KL]). Given a representation ρ : G K → U(n) (resp. O(n)), we have ∆K,ρ (t) ≡ ∆K,ρ (t) mod ηtl η∈R× ,l∈Z . 4 T. Kitayama 3 Sign-determined Reidemeister torsion In this section, we review the deﬁnition of Turaev’s sign-determined Reidemeister torsion. See [T1], [T2] for more details. For two bases u and v of an n-dimensional vector space over a ﬁeld F, [u/v] denotes the determinant of the base change matrix from v to u. ∂n ∂1 Let C∗ = (0 → Cn − Cn−1 → · · · − C0 → 0) be a chain complex of ﬁnite dimensional → → vector spaces over F. For given bases bi of Im ∂i+1 and hi of Hi (C∗ ), we can choose a basis bi ∪ hi ∪ bi−1 of Ci as follows. First, we choose a lift hi of hi in Ci+1 and obtain a basis bi ∪ hi of ˜ ˜ ˜ ˜ Ker ∂i . Consider the exact sequence 0 → Im ∂i+1 → Ker ∂i → Hi (C∗ ) → 0. Then we choose a lift bi−1 of bi−1 in Ci and obtain a basis (bi ∪ hi ) ∪ bi−1 of Ci . Consider the ˜ ˜ ˜ exact sequence 0 → Ker ∂i → Ci → Im ∂i → 0. Deﬁnition 3.1. For given bases c = (ci ) of C∗ and h = (hi ) of H∗ (C∗ ), we choose a basis b = (bi ) of Im ∂∗ and deﬁne n Tor(C∗ , c, h) := (−1)|C∗ | ∈ F ×, i+1 [bi ∪ hi ∪ bi−1 /ci ](−1) ˜ ˜ i=0 where n j j |C∗ | := ( dim Ci )( dim Hi (C∗ )). j=0 i=0 i=0 Remark 3.2. It can be easily checked that Tor(C∗ , c, h) does not depend on the choices of b, bi ˜ ˜ and hi . Now let us apply the above algebraic torsion to the geometric situations. Let X be a con- nected ﬁnite CW-complex. By a homology orientation of X we mean an orientation of the homology group H∗ (X; R) = H (X; R) as a real vector space. i i Deﬁnition 3.3. For a representation ρ : π1 X → GLn (F) such that the twisted homology group ⊕n H∗ (X; Fρ ) vanishes and a homology orientation o, we deﬁne the sign-determined Reidemeister torsion T ρ (X, o) of ρ and o as follows. We choose a lift ei of each cell ei in X and bases h of ˜ H∗ (X; R) which is positively oriented with respect to o and f1 , . . . , fn of F ⊕n . Then, T ρ (X, o) := τn Tor(C∗ (X) ⊗ρ F ⊕n , c) ∈ F × , 0 ˜ where τ0 := sgn Tor(C∗ (X; R), c, h), c := e1 , . . . , edimC∗ , c := e1 ⊗ f1 , . . . , e1 ⊗ fn , . . . , edimC∗ ⊗ f1 , . . . , edimC∗ ⊗ fn . ˜ ˜ ˜ ˜ ˜ Normalization of twisted Alexander invariants 5 Remark 3.4. It is known that T ρ (X, o) does not depend on the choice of ei , h and f1 , . . . , fn and ˜ is well-deﬁned as a simple homotopy invariant up to multiplication of an element in Im(det ◦ρ). See [T1]. Here let us consider the knot exterior E K . In this case, we can equip E K with its canonical homology orientation ωK as follows. We have H∗ (E K ; R) = H0 (E K ; R) ⊕ t and deﬁne ωK := [ [pt], t ], where [pt] is the homology class of a point. Deﬁnition 3.5. For a representation ρ : G K → GLn (F) such that the twisted homology group H∗ (X; F(t)⊕n ) vanishes, the sign-determined Reidemeister torsion T K,ρ (t) of ρ is deﬁned by α⊗ρ T α⊗ρ (E K , ωK ). Here we consider α ⊗ ρ : G K → GLn (F[t, t−1 ]) → GLn (F(t)). In the later section, we generalize the following theorem. Theorem 3.6 ([Ki], [KL]). For a representation ρ : G K → GLn (F) such that the twisted homol- ogy group H∗ (X; F(t)⊕n ) vanishes, we have α⊗ρ ∆K,ρ (t) ≡ T K,ρ (t) mod ηtl η∈F × ,l∈Z . 4 Construction Now we establish one of our main results. We get rid of the multiplicative ambiguity of twisted Alexander invariants via a combinatorial method. For f (t) = p(t)/q(t) ∈ Q(R)(t) (p, q ∈ R[t, t−1 ]), we deﬁne deg f := deg p − deg q, h-deg f := (the highest degree of p) − (the highest degree of q), l-deg f := (the lowest degree of p) − (the lowest degree of q), (the coeﬃcient of the highest degree term of p) c( f ) := . (the coeﬃcient of the highest degree term of q) We make use of a combinatorial group theoretical approach constructed by Wada in [Wad]. Deﬁnition 4.1. Given a ﬁnite presentable group G = x1 , . . . , xm | r1 , . . . , rn , the operations of the following types for any word w in x1 , . . . , xm , are called the strong Tietze transformations: Ia. To replace one of the relators ri by its inverse ri−1 . Ib. To replace one of the relators ri by its conjugate wri w−1 . Ic. To replace one of the relators ri by ri r j for any j i. II. To add a new generator y and a new relator yw−1 . (Namely, the resulting presentation is x1 , . . . , xm , y | r1 , . . . , rn , yw−1 .) If a presentation is transformable to another by a ﬁnite sequence of operations of above types and their inverse operations, we say that the two presentations are strongly Tietze equivalent. 6 T. Kitayama Remark 4.2. The deﬁciency of G does not change via the strong Tietze transformations. Wada shows the following lemma. Lemma 4.3 ([Wad]). All the Wirtinger presentations of a given link in S 3 are strongly Tietze equivalent to each other. Let ϕ : Z[G K ] → Z be the augmentation homomorphism. (Namely, ϕ(γ) = 1 for any element γ of G K .) For a ﬁxed presentation x1 , . . . , xm | r1 , . . . , rm−1 of G K , we denote Aϕ,k and Aα,k by ˜ ∂ri ∂ri ϕ ∂x j and α ∂x j ˜ as in Section 2. j k j k We eliminate the ambiguity of ηtl in Deﬁnition 2.1 as follows. Deﬁnition 4.4. Given a representation ρ : G K → GLn (R), we choose a presentation x1 , . . . , xm | r1 , . . . , rm−1 of G K which is strongly Tietze equivalent to a Wirtinger presenta- tion and an index 1 ≤ k ≤ m such that h-deg α(xk ) 0. Then we deﬁne the normalized twisted Alexander invariant associated to ρ as δn det AΦ,k 1 1 ∆K,ρ := n d ∈ Q(R)( 2 )(t 2 ), ( t ) det Φ(xk − 1) where := det ρ(µ), δ := sgn(h-deg α(xk ) det Aϕ,k ), 1 d := (h-deg det Aα,k + l-deg det Aα,k − h-deg α(xk )). ˜ ˜ 2 Theorem 4.5. ∆K,ρ is an invariant of a linear representation ρ. Proof. From Lemma 4.3, we have to check (i) the independence of the choice of k and (ii) the invariance for each operation of Deﬁnition 4.1. We assume that we can choose another index k also satisfying the condition h-deg α(xk ) 0. We set δ := sgn(h-deg α(xk ) det Aϕ,k ), 1 d := (h-deg det Aα,k + l-deg det Aα,k − h-deg α(xk )). ˜ ˜ 2 Since m ∂ri (x j − 1) = ri − 1, j=1 ∂x j we have ∂ri det AΦ,k det Φ(xk − 1) = det . . . , Φ Φ(xk − 1), . . . , ∂xk ∂ri = det . . . , − Φ Φ(x j − 1), . . . , j k ∂x j ∂ri = det . . . , −Φ Φ(xk − 1), . . . , ∂xk = (−1)n(k−k ) det AΦ,k det Φ(xk − 1). Normalization of twisted Alexander invariants 7 Similarly, we obtain det Aα,k det α(xk − 1) = (−1)k−k det Aα,k det α(xk − 1). ˜ ˜ ˜ ˜ Hence d = d. Moreover, by dividing this equality by (t − 1) and taking t → 1, we can see that h-deg α(xk ) det Aϕ,k = (−1)k−k h-deg α(xk ) det Aϕ,k . Hence δ = (−1)k−k δ. This concludes the proof of (i). Next, we consider the strong Tietze transformations. Since ∂(ri−1 ) ∂ri = −ri , ∂x j ∂x j ∂(wri w−1 ) ∂ri =w , ∂x j ∂x j ∂(ri rl ) ∂ri ∂rl = + ri , ∂x j ∂x j ∂x j the changes of each value by the transformation Ia, Ib and Ic are as follows. By the transfor- mation Ia, det AΦ,k → (−1)n det AΦ,k , δ → −δ and d does not change. By the transformation Ib, det AΦ,k → ( tn )deg α(w) det AΦ,k , δ does not change and d → d + deg α(w). By the transformation Ic and II, it is easy to see that all the values do not change. This concludes the proof of (ii). From the construction, the following lemma holds. Lemma 4.6. (i)For a representation ρ : G K → GLn (R), 1 l ∆K,ρ (t) ≡ ∆K,ρ (t) mod 2 , ηt 2 η∈R× ,l∈Z . (ii)If ρ is trivial (i.e., Φ = α), ˜ − t− 2 ) = (t 2 − t− 2 )∆K,ρ (t), 1 1 1 1 K (t 2 where K (z) is the Conway polynomial of K. Proof. Since (i) is clear from Theorem 2.3 and the deﬁnition, we prove (ii). We set f (t) = (t 2 − t− 2 )∆K,ρ (t). 1 1 It is easy to see that f (t) ≡ ∆K (t) mod ±t . Moreover, we can check that f (1) = 1, h-deg f + l-deg f = 0, which establishes the formula. 8 T. Kitayama 5 Relation to sign-determined Reidemeister torsion In this section, we generalize Theorem 2.5 and Theorem 3.6. Here we only consider the case that R is a ﬁeld F. First, we also normalize sign-determined Reidemeister torsion as twisted Alexander invari- ants. Deﬁnition 5.1. For a representation ρ : G K → GLn (F) such that the twisted homology group H∗ (E K ; F(t)⊕n ) vanishes, we deﬁne T K,ρ (t) as follows. We choose a lift ei in E K of each cell ei , α⊗ρ ˜ bases h of H∗ (E K ; R) which is positively oriented with respect to ωK and f1 , . . . , fn of F(t)⊕n . Then τn T K,ρ (t) := n0 d Tor(C∗ (E K ) ⊗α⊗ρ F(t)⊕n , c) ∈ F(t)× , ˜ ( t) where := det ρ(µ), τ0 := sgn Tor(C∗ (E K ; R), c, h), 1 d := (h-deg Tor(C∗ (E K ) ⊗α Q(t), c0 ) + l-deg Tor(C∗ (E K ) ⊗α Q(t), c0 )), ˜ ˜ 2 c := e1 , . . . , edimC∗ , c0 := e1 ⊗ 1, . . . , edimC∗ ⊗ 1 , ˜ ˜ ˜ c := e1 ⊗ f1 , . . . , e1 ⊗ fn , . . . , edimC∗ ⊗ f1 , . . . , edimC∗ ⊗ fn . ˜ ˜ ˜ ˜ ˜ Remark 5.2. We can also deﬁne normalized Reidemeister torsion for a link by a similar method. One can prove the following lemma by a similar way as in the non-normalized case. As a reference, see [T1]. Lemma 5.3. T K,ρ is invariant under homology orientation preserving simple homotopy equiva- lence. Remark 5.4. From the result of Waldhausen [Wal], the Whitehead group Wh(G K ) is trivial for a knot group G K in general. Therefore homotopy equivalence between ﬁnite CW-complexes whose fundamental groups are knot groups is simple homotopy equivalence. Let F be a ﬁeld with (possibly trivial) involution f → f¯. We extend the involution to F(t) by taking t → t−1 . We equip F(t)⊕n with the standard hermitian inner product (·, ·) deﬁned by (v, w) := t vw, ¯ where v, w ∈ F(t)⊕n and t v is the transpose of v. For a representation ρ : G K → GLn (F), we deﬁne a representation ρ† : G K → GLn (F) by ρ† (γ) := ρ(γ−1 )∗ , where γ ∈ G K and A∗ := t A for a matrix A. We can also reﬁne the duality theorem for sign-determined Reidemeister torsion as follows. Normalization of twisted Alexander invariants 9 Theorem 5.5. If the twisted homology group H∗ (E K ; F(t)⊕n ) vanishes for a representation α⊗ρ ρ : G K → GLn (F), then so does H∗ (E K ; F(t)⊕n † ) and we have α⊗ρ T K,ρ† (t) = (−1)n T K,ρ (t). The proof is based on the following observation. Let (E K , {ei }) denote the PL manifold E K with the dual cell structure and choose a lift ei which is the dual of ei . In the remainder of this ˜ ˜ section, for abbreviation, we write Cq := Cq (E K ) ⊗α Q(t), Cρ,q := Cq (E K ) ⊗α⊗ρ F(t)⊕n , Cq := Cq (∂E K ) ⊗α Q(t), Cρ,q := Cq (∂E K ) ⊗α⊗ρ F(t)⊕n , Cq := Cq (E K , ∂E K ) ⊗α Q(t), Cρ,q := Cq (E K , ∂E K ) ⊗α⊗ρ F(t)⊕n , Dq := Cq (E K ) ⊗α Q(t), Dρ,q := Cq (E K ) ⊗α⊗ρ† F(t)⊕n , Bq := Im(∂ : Cq+1 → Cq ), Bρ,q := Im(∂ : Cρ,q+1 → Cρ,q ), Bq := Im(∂ : Cq+1 → Cq ), Bρ,q := Im(∂ : Cρ,q+1 → Cρ,q ). Note that since direct computation gives H∗ (∂E K ; F(t)⊕n ) = 0 α⊗ρ (5.1) (See, for example, [KL, Subsection 3.3.].), we have i dim Bρ,i = (−1)i− j dim Cρ, j j=0 i (5.2) = (−1)i− j n dim C j = n dim Bi j=0 Similarly, if H∗ (E K ; F(t)⊕n ) = 0, then from (5.1) and the long exact sequence of the pair α⊗ρ (E K , ∂E K ), H∗ (E K , ∂E K ; F(t)⊕n ) = 0 and so α⊗ρ dim Bρ,i = n dim Bi . (5.3) The well known inner product [·, ·] : Cq (E K ) × C3−q (E K , ∂E K ) → Z[G K ] (See, for example, [M, Lemma 2.].) deﬁned by [˜ i , e j ] := e ˜ (˜ i , e j · γ−1 )γ, e ˜ γ∈G K where (·, ·) denote the intersection number, induces an inner product ·, · : Dρ,q × Cρ,3−q → C(t) 10 T. Kitayama deﬁned by ei ⊗ v, e j ⊗ w := (v, [˜ i , e j ] · w), ˜ ˜ e ˜ where v, w ∈ C(t)⊕n . We see at once that this is well-deﬁned. Thus Dρ,q (Cρ,3−q )∗ . (5.4) The diﬀerential ∂q of Dρ,q corresponds with (−1)q ∂∗ of (Cρ,3−q )∗ under this isomorphism. Sim- 3−q ilarly, we have Dq (C3−q )∗ . (5.5) Lemma 5.6. For any representation ρ : G K → GLn (F), Hq (E K ; F(t)⊕n † ) α⊗ρ H3−q (E K ; F(t)⊕n )∗ . α⊗ρ Proof. From (5.4) and the universal coeﬃcient theorem, we can see that Hq (E K ; F(t)⊕n † ) α⊗ρ H3−q (E K , ∂E K ; F(t)⊕n )∗ . α⊗ρ From (5.1) and the long exact sequence of the pair (E K , ∂E K ), H∗ (E K ; F(t)⊕n ) α⊗ρ H∗ (E K , ∂E K ; F(t)⊕n ). α⊗ρ This completes the proof. Now we prove the theorem. Proof of Theorem 5.5. Lemma 5.6 gives the ﬁrst assertion. We use the notation of Deﬁnition 5.1. We choose an orthonormal basis f1 , . . . , fn of F(t)⊕n with respect to the hermitian product (·, ·) deﬁned above. Let c , c , c0 , c0 , c and c be induced bases of C∗ (∂E K ), C∗ (E K , ∂E K ), C∗ , ˜ ˜ ˜ ˜ C∗ , Cρ,∗ and Cρ,∗ by c, c0 and c. We set c∗ := e1 , . . . , edimC∗ , c∗ := e1 ⊗ 1, . . . , edimC∗ ⊗ 1 , ˜0 c∗ := e1 ⊗ f1 , . . . , e1 ⊗ fn , . . . , edimC∗ ⊗ f1 , . . . , edimC∗ ⊗ fn . ˜ ˜ ˜ ˜ ˜ From (5.4) and the duality for algebraic torsion ([T2, Theorem 1.9]), Tor(Dρ,∗ , c∗ ) = (−1) ˜ i dim Bρ,i−1 dim Bρ,i Tor(Cρ,∗ , c ). ˜ On the other hand, from the exact sequence 0 → Cρ,∗ → Cρ,∗ → Cρ,∗ → 0 and the multiplicativity for algebraic torsion ([T2, Theorem 1.5]), Tor(Cρ,∗ , c) = (−1) ˜ i dim Bρ,i−1 dim Bρ,i Tor(Cρ,∗ , c ) Tor(Cρ,∗ , c ). ˜ ˜ Normalization of twisted Alexander invariants 11 Therefore, we obtain i (dim Bρ,i−1 +dim Bρ,i−1 ) dim Bρ,i Tor(Cρ,∗ , c) = (−1) ˜ Tor(Cρ,∗ , c )Tor(Dρ,∗ , c∗ ). ˜ ˜ (5.6) Similarly, i (dim Bi−1 +dim Bi−1 ) dim Bi Tor(C∗ , c0 ) = (−1) ˜ Tor(C∗ , c0 )Tor(D∗ , c∗ ). ˜ ˜0 (5.7) We set 1 d := (h-deg Tor(C∗ , c0 ) + l-deg Tor(C∗ , c0 )), ˜ ˜ 2 1 d∗ := (h-deg Tor(D∗ , c∗ ) + l-deg Tor(D∗ , c∗ )). ˜0 ˜0 2 From (5.7), we have d = d − d∗ . (5.8) From Lemma 4.6(ii) and Theorem 5.7, lim τ0 (t 2 − t− 2 ) Tor(C∗ , c0 ) = − 1 1 ˜ K (0) t→1 = −1. Similarly, lim τ∗ (t 2 − t− 2 ) Tor(D∗ , c∗ ) = −1, 1 1 0 ˜0 t→1 where τ∗ := sgn Tor(C∗ (E K ; R), c∗ , h). 0 By multiply (5.7) by (t 2 − t− 2 ) and taking t → 1, we obtain 1 1 i (dim Bi−1 +dim Bi−1 ) dim Bi τ0 = −(−1) τ0 τ∗ , 0 (5.9) where τ0 := lim Tor(C∗ , c0 ). ˜ t→1 From (5.2), (5.3), (5.6), (5.8) and (5.9), τn T K,ρ (t) = 0 Tor(Cρ,∗ , c) ˜ ( tn )d (τ0 )n (τ∗ )n = (−1) n n )d Tor(Cρ,∗ , c ) · n d∗ Tor(Dρ,∗ , c∗ ). ˜ 0 ˜ ( t ( t) Direct computation gives Tor(C∗ , c0 ) = τ0 td . ˜ (See, for example, [KL, Subsection 3.3.].) Since the normalized invariants do not change via conjugation of representations, we can assume ρ(µ) and ρ(λ) are diagonal matrices. This de- duces Tor(Cρ,∗ , c ) = (τ0 )n ( tn )d . ˜ 12 T. Kitayama Thus (τ0 )n Tor(Cρ,∗ , c ) = 1. ˜ ( tn )d It can be easily seen that (τ∗ )n 0 Tor(Dρ,∗ , c∗ ) = T K,ρ† (t). ˜ ( tn )d∗ This proves the theorem. In the normalized setting, Theorem 3.6 also holds. Theorem 5.7. For a representation ρ : G K → GLn (F) such that the twisted homology group H∗ (E K ; F(t)⊕n ) vanishes, we have α⊗ρ ∆K,ρ (t) = T K,ρ (t). Proof. We choose a Wirtinger presentation G K = x1 , . . . , xm | r1 , . . . , rm−1 and take the CW- complex W corresponding with the presentation. Namely, W has one vertex, m edges and (m − 1) 2-cells attached by the relations r1 , . . . , rm−1 . Let the words x1 , . . . , xm and r1 , . . . , rm−1 also denote the cells. It is easy to see that the exterior E K collapses to W. This implies that W is simple homotopy equivalent to E K from Remark 5.4. Thus we can compute the normalized torsion T K,ρ as that of W from Lemma 5.3. C∗ (W; R) is m−1 m ∂2 ∂1 0 → → Rr j − → Rxi − Rpt → 0, j=1 i=1 where ∂1 = 0, ∂r j ∂2 = ϕ . ∂xi Let c0 = pt, c1 = x1 , . . . , xm and c2 = r1 , . . . , rm−1 . We choose b1 = ∂c2 and h0 = [pt], h1 = [xk ] (1 ≤ k ≤ m). Then [b1 ∪ h1 /c1 ] ˜ τ0 = sgn(−1)|C∗ (W;R)| [h0 /c0 ][b1 /c2 ] ˜ ˜ 0 . . . 0 ϕ 1 ∂r j = − sgn det ∂xi 0 . . . 0 = (−1)k+m+1 δ. Normalization of twisted Alexander invariants 13 We deﬁne an involution ¯ : Z[G K ] → Z[G K ] by extending the inverse operation γ → γ−1 of · G K linearly. We can choose lifts pt, xi and r j such that C∗ (W) ⊗α⊗ρ F(t)⊕n is ˜ ˜ ∂2 ˜ ∂1 ˜ 0 → → F(t)(˜ j ⊗ fl ) − r → F(t)( xi ⊗ fl ) − ˜ F(t)( pt ⊗ fl ) → 0, 1≤ j≤m−1,1≤l≤n 1≤i≤m,1≤l≤n 1≤l≤n where ∂1 ( xi ⊗ fl ) = pt ⊗ Φ( xi − 1) fl ˜ ˜ ˜ m ∂r j ∂2 (˜ j ⊗ fl ) = ˜ r xi ⊗ Φ fl . ˜ ∂x i=1 i Let c0 = pt ⊗ f1 , . . . , pt ⊗ fn , c1 = x1 ⊗ f1 , . . . , x1 ⊗ fn , . . . , xm ⊗ f1 , . . . , xm ⊗ fn and c2 = ˜ ˜ ˜ ˜ r1 ⊗ f1 , . . . , r1 ⊗ fn , . . . , rm−1 ⊗ f1 , . . . , rm−1 ⊗ fn . We choose b0 = ∂ xk ⊗ f1 , . . . , xk ⊗ fn and ˜ ˜ ˜ ˜ ˜ ˜ ⊕n b1 = ∂c2 . Since the twisted homology group H∗ (W; F(t)α⊗ρ ) vanishes, |C∗ (W) ⊗α⊗ρ F(t)⊕n | = 0 and so [b1 ∪ b0 /c1 ] ˜ Tor(C∗ (W) ⊗α⊗ρ F(t)⊕n , c0 , c1 , c2 ) = ˜ ˜ ˜ [b0 /c0 ][b1 /c2 ] ˜ 0 . . . 0 det Φ ∂xij I ∂r 0 . . . 0 = det Φ(xk − 1) ∂ri det t Φ ∂x j = (−1)n(k+m) . det t Φ(xk − 1) Similarly, we have ∂ri det α ˜ ∂x j Tor(C∗ (W) ⊗α Q(t), c0 , c1 , c2 ) = (−1)(k+m) ˜ ˜ ˜ . det α(xk − 1) ˜ Hence d = −d and so T K,ρ (t) = (−1)n ∆K,ρ† (t), where we consider the trivial involution on F. From Theorem 5.5, we obtain the desired for- mula. From the above theorems and the following lemma, we have the duality theorem for nor- malized twisted Alexander invariants. 14 T. Kitayama Lemma 5.8. If H∗ (E K ; F(t)⊕n ) does not vanish, then we have α⊕ρ ∆K,ρ (t) = ∆K,ρ† (t) = 0. Proof. If H∗ (E K ; F(t)⊕n ) does not vanish, then neither does H∗ (E K ; F(t)⊕n † ) from Lemma 5.6. α⊗ρ α⊗ρ Since 2 dim Hq (E K ; F(t)⊕n ) = nχ(E K ) α⊕ρ q=0 = 0, from the assumption and (5.1), we have H1 (E K ; F(t)⊕n ) α⊗ρ 0 and so ∆K,ρ (t) = 0. Similarly, we obtain ∆K,ρ† (t) = 0, which proves the lemma. Theorem 5.9. Given a representation ρ : G K → GLn (F), we have ∆K,ρ† (t) = (−1)n ∆K,ρ (t). For a unitary representation ρ, the diﬀerence between the highest coeﬃcient of ∆K,ρ (t) and the lowest coeﬃcient of it is not clear from Theorem 2.5 because of the ambiguity. However, this diﬀerence is strictly determined from the following corollary. Corollary 5.10. For a representation ρ : G K → U(n) or O(n), we have ∆K,ρ (t) = (−1)n ∆K,ρ (t). Example 5.11. Let K be the (p, q) torus knot (p, q > 1 and (p, q) = 1). It is well known that the knot group has a presentation G K = x, y | x p y−q , where h-deg α(x) = q and h-deg α(y) = p. The 2-dimensional complex W corresponding with this presentation is K(G K , 1). Therefore we can use this presentation for the computation via Lemma 5.3, Remark 5.4 and Theorem 5.7. From the result of Klassen [Kl], all the irreducible S U(2)-representations up to conjugation are given as follows: ρa,b,s : G K → S U(2) : cos aπ + i sin aπ 0 x→ p p , 0 cos p − i sin aπ aπ p cos bπ + i sin bπ cos πs sin bπ sin πs y→ q q q , − sin bπ sin πs q cos bπ − i sin bπ cos πs q q where a, b ∈ N, 1 ≤ a ≤ p − 1, 1 ≤ b ≤ q − 1, a ≡ b mod 2 and 0 < s < 1. The normalized twisted Alexander invariants of the torus knot for these representations are as follows: pq pq (t 2 − (−1)a t− 2 )2 ∆K,ρa,b,s (t) = p . (t − 2 cos bπ + t−p )(tq − 2 cos aπ + t−q ) q p Normalization of twisted Alexander invariants 15 6 Applications Now we consider applications of the normalized invariants. First we generalize the result of Goda-Kitano-Morifuji and Friedl-Kim. We denote by g(K) the genus of K. Their results are as follows. Theorem 6.1 ([GKM]). For a ﬁbered knot K and a unimodular representation ρ : G K → S L2n (F), c(∆K,ρ ) is well-deﬁned and is 1. Theorem 6.2 ([C],[FK]). For a ﬁbered knot K and a representation ρ : G K → GLn (R), ∆1 is K,ρ a monic polynomial and deg ∆K,ρ = n(2g(K) − 1), where “monic” means that the highest and lowest coeﬃcients of a polynomial are units. In the normalized setting, we have the following theorem. Theorem 6.3. For a ﬁbered knot K and a representation ρ : G K → GLn (R), deg ∆K,ρ = 2 h-deg ∆K,ρ = n(2g(K) − 1), n g(K)− 1 c(∆K,ρ ) = c( K) 2. Proof. The equality deg ∆K,ρ = n(2g(K) − 1) can be obtained from Theorem 6.2. Since we have ∆K,i◦ρ = ∆K,ρ , where i is the natural inclusion GLn (R) → GLn (Q(R)), we can assume R is a ﬁeld F. Let ψ denote the automorphism of a surface group induced by the monodromy map. We can take the following presentation of the knot group by using the ﬁbered structure: x1 , . . . , x2g , h | ri := hxi h−1 ψ∗ (xi )−1 , 1 ≤ i ≤ 2g(K) where α(xi ) = 1 for all i and α(h) = t. It is easy to see that the corresponding CW-complex is homotopy equivalent to the exterior E K . Thus we can compute the invariant by using the presentation as in Example 5.11. Since ∂ri h − ∂xi i i = j ∂ψ∗ (x ) = ∂ψ∗ (xi ) , ∂x j − ∂x j i j we have det Aα,2g(K)+1 = t2g(K) + · · · + 1, ˜ ∂ψ∗ (xi ) det AΦ,2g+1 = 2g(K) 2ng(K) t + · · · + (−1)n det(Φ( )), ∂x j det Φ(h − 1) = tn + · · · + (−1)n . From the classical theorem of Neuwirth which states that the degree of the Alexander polyno- mial of a ﬁbered knot equals the twice genus, we can determine that the lowest degree term of 16 T. Kitayama the ﬁrst equality is 1. Since − t− 2 ) 1 1 δ = sgn c( K) K (t 2 t=1 = c( K) 1 d = g(K) − , 2 h-deg ∆K,ρ = n(g(K) − 1 ) and c(∆K,ρ ) = c( 2 K) n 2g(K)−1 . Next we study a behavior of the highest degree of a normalized invariant. Deﬁnition 6.4. A Seifert surface for a knot K is said to be canonical if it is obtained from a diagram of K by applying the Seifert algorithm. The minimum genus over all canonical Seifert surfaces is called the canonical genus and denoted by gc (K). A Seifert surface S is said to be free if π1 (S 3 \ S ) is a free group. This condition is equivalent to that S 3 \ N(S ) is a handlebody, where N(S ) is an open regular neighborhood of S . The minimum genus over all free Seifert surfaces is called the free genus and denoted by g f (K). Remark 6.5. Since every canonical Seifert surfaces is free, we have the following fundamental inequality: g(K) ≤ g f (K) ≤ gc (K). We obtain an estimate of free genus from below via the highest degree of the invariants. Theorem 6.6. For a representation ρ : G K → GLn (R), the following inequality holds: 2 h-deg ∆K,ρ ≤ n(2g f (K) − 1). Proof. We choose a free Seifert surface S with genus g f (K) and take a bicollar S × [−1, 1] of S such that S × 0 = S . Let ι± : S → S 3 \ S be the embeddings whose images are S × {±1}. Picking generator sets {a1 , . . . , a2g f (K) } of π1 S and {x1 , . . . , x2g f (K) } of π1 (S 3 \ S ) and setting ui := (ι+ )∗ (ai ) and vi := (ι− )∗ (ai ) for all i, we have a presentation x1 , . . . , x2g f (K) , h | ri := hui h−1 v−1 , 1 ≤ i ≤ 2g f (K) i of G K where α(xi ) = 1 for all i and α(h) = t. Collapsing surfaces S × ∗ and the handlebody S 3 \ (S × [−1, 1]) to bouquets, we can realize the 2-dimensional complex corresponding with this presentation as a deformation retract of E K . Therefore we can compute the invariant by using the presentation as in Example 5.11. Since ∂ri ∂ui ∂vi =h − , ∂x j ∂x j ∂x j we have h-deg ∆K,ρ = h-deg det AΦ,2g f (K)+1 − nd − n ≤ 2ng f (K) − nd − n. Normalization of twisted Alexander invariants 17 The proof is completed by showing that d = g f (K) − 1 . 2 Let V be the Seifert matrix with respect to the basis [a1 ], . . . , [a2g f (K) ] ∈ H1 (S ; Z) and [a1 ]∗ , . . . , [a2g f (K) ]∗ ∈ H1 (S 3 \ S ; Z) the dual basis, i. e. lk([ai ], [a j ]∗ ) = δi, j . We denote by A± the matrices representing (ι± )∗ : H1 (S ; Z) → H1 (S 3 \S ; Z) with respect to the bases [a1 ], . . . , [a2g f (K) ] and [x1 ], . . . , [x2g f (K) ] and by P the base change matrix of H1 (S 3 \ S ; Z) from [x1 ], . . . , [x2g f (K) ] to [a1 ]∗ , . . . , [a2g f (K) ]∗ . It is well known that the matrices representing (ι+ )∗ and (ι− )∗ : H1 (S ; Z) → H1 (S 3 \ S ; Z) with respect to the bases [a1 ], . . . , [a2g f (K) ] and [a1 ]∗ , . . . , [a2g f (K) ]∗ are V and t V. Hence det Aα,2g f (K)+1 = det(tt A+ − t A− ) ˜ = det(tA+ − A− ) = det(tPV − Pt V) = ± det(tV − t V), and d = g f (K) − 1 2 as required. Example 6.7. Let K be the knot 11n 73 illustrated in Figure 1. The normalized Alexander poly- nomial of K is t2 − 2t + 3 − 2t−1 + t−2 . Figure 1 The Wirtinger presentation of the diagram in Figure 1 consists of 11 generators and 10 relations: −1 −1 −1 −1 x5 x1 x5 x2 , x11 x2 x11 x3 , −1 −1 −1 −1 x9 x4 x9 x3 , x7 x5 x7 x4 , −1 −1 −1 −1 x1 x5 x1 x6 , x8 x7 x8 x6 , −1 −1 −1 −1 x5 x8 x5 x7 , x10 x9 x10 x8 , −1 −1 −1 −1 x4 x10 x4 x9 , x2 x10 x2 x11 . 18 T. Kitayama Let ρ : G K → S L2 (F2 ) be a nonabelian representation over F2 deﬁned as follows: 1 0 1 1 , if i = 4, 8 0 1 ρ(xi ) = 1 0 , if i = 7, 9 . 1 1 0 1 , otherwise From them, We have the following: ∆K,ρ (t) = t5 + t + t−1 + t−5 . Since deg ∆K,ρ 2(deg ∆K − 1), K is not ﬁbered. Moreover, from Proposition 6.6, 10 ≤ 2(2g f (K) − 1). Therefore g f (K) ≥ 3. On the other hand, we obtain a canonical Seifert surface with genus 3 by applying the Seifert algorithm to the diagram in Figure 1. Thus g f (K) ≤ gc (K) ≤ 3. By these inequalities we conclude g f (K) = gc (K) = 3. Remark 6.8. 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Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan E-mail address: kitayama@ms.u-tokyo.ac.jp