UTMS 2004–36

November 16, 2004

Twisted Alexander polynomial
and surjectivity of a group homomorphism
byTeruaki Kitano, Masaaki Suzuki
and Masaaki WadaÌ

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANTWISTED ALEXANDER POLYNOMIAL AND SURJECTIVITY
OF A GROUP HOMOMORPHISMTERUAKI KITANO, MASAAKI SUZUKI AND MASAAKI WADA
Abstract. If ϕ : G → G′ is a surjective homomorphism, we prove that thetwisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G′ . As an application, we show non-existence of surjective homomorphism between certain knot groups.

1. Introduction
Suppose that G is a finitely presentable group with a surjective homomorphism
to the free abelian group of rank l, e.g., abelianization. Let ρ : G → GL(n; R) be a
linear representation. The twisted Alexander polynomial of G associated to ρ wasintroduced in [8] and is defined to be a rational expression of l indeterminates.
Let ϕ : G → G′ be a surjective homomorphism. Each representation ρ′ : G′ →GL(n; R) naturally induces a representation of G, namely, ρ = ρ′ ◦ ϕ. In this paper
we prove the following:
Main theorem . The twisted Alexander polynomial of G associated to ρ is divisible by the twisted Alexander polynomial of G′ associated to ρ′ .
The corresponding fact about the (classical) Alexander polynomial is well known[1].
We present two separate proofs of the main theorem. First we give a purely
algebraic proof in §3. If G is a knot group, the twisted Alexander polynomial of G
may be regarded as the Reidemeister torsion. In §4, we provide another proof ofthe main theorem in case when G and G′ are knot groups, from the view point ofReidemeister torsion.
In the last section, we show non-existence of surjective homomorphism between
certain knot groups, as an application of the main theorem.
2. Twisted Alexander polynomial
In this section, we recall briefly the definition of the twisted Alexander polynomial.
Let G be a finitely presentable group. Choose and fix a presentation as follows:
G = hx1 , . . . , xu | r1 , . . . , rv i.
1991 Mathematics Subject Classification. Primary 57M25, Secondary 57M05.
Key words and phrases. twisted Alexander polynomial, Reidemeister torsion.
The first author is supported in part by Grand-in-Aid for Scientific Research (No. 14740037),
The Ministry of Education, Culture, Sports, Science and Technology, Japan. The second author
is supported by the 21 century COE program at Graduate School of Mathematical Sciences, theUniversity of Tokyo.
1

2

TERUAKI KITANO, MASAAKI SUZUKI AND MASAAKI WADA

We denote by α : G → Zl a surjective homomorphism to the free abelian groupwith generators t1 , . . . , tl and ρ : G → GL(n; R) a linear representation, where
R is a unique factorization domain. These maps naturally induce ring homomorphisms ρ̃ and α̃ from Z[G] to M (n; R) and Z[t1 ±1 , . . . , tl ±1 ] respectively, where
M (n; R) denotes the matrix algebra of degree n over R. Then ρ̃ ⊗ α̃ defines a ringhomomorphism
Z[G] → M n; R[t1 ±1 , . . . , tl ±1 ] .
Let Fu be the free group on generators x1 , . . . , xu andΦ : Z[Fu ] → M n; R[t1 ±1 , . . . , tl ±1 ]the composite of the surjection Z[Fu ] → Z[G] induced by the fixed presentationand the map ρ̃ ⊗ α̃ : Z[G] → M (n; R[t1 ±1 , . . . , tl ±1 ]).
We define the v × u matrix M whose (i, j) component is the n × n matrix



∂ri
∈ M n; R[t1 ±1 , . . . , tl ±1 ] ,Φ∂xjwhere ∂/∂x denotes the Fox derivation. This matrix M is called the Alexander
matrix of the presentation of G associated to the representation ρ.
It is easy to see that there is an integer 1 ≤ j ≤ u such that det Φ(xj − 1) 6= 0.
For such j, let us denote by Mj the v ×(u−1) matrix obtained from M by removing
the j-th column. We regard Mj as an nv × n(u − 1) matrix with coefficients inR[t1 ±1 , . . . , tl ±1 ]. Moreover, for an n(u − 1)-tuple of indices
I = i1 , i2 , . . . , in(u−1) ,1 ≤ i1 < i2 < · · · < in(u−1) ≤ nv

we denote by MjI the n(u − 1) × n(u − 1) square matrix consisting of the ik -th rowof the matrix Mj , where k = 1, 2, . . . , n(u − 1).
Then the twisted Alexander polynomial (see [8]) of a finitely presented group G
for a representation ρ : G → GL(n; R) is defined to be a rational expression∆G,ρ (t1 , . . . , tl ) =gcdI (det MjI )det Φ(xj − 1)and moreover well-defined up to a factor ǫt1 ε1 · · · tl εl , where ǫ ∈ R× , εi ∈ Z. See[8], [6], [2] and [3] for more precise definition and applications.
3. Main theorem and the algebraic proof
In this section, we prove the following main theorem of this paper.
Theorem 3.1. Let G and G′ be finitely presentable groups and α, α′ surjective
homomorphisms from G, G′ to Zl respectively. Suppose that there exists a surjective
homomorphism ϕ : G → G′ such that α = α′ ◦ ϕ. Then ∆G,ρ is divisible by ∆G′ ,ρ′for any representation ρ′ : G′ → GL(n; R), where ρ = ρ′ ◦ ϕ. That is to say, thequotient of ∆G,ρ by ∆G′ ,ρ′ is a genuine polynomial.
Proof. Choose and fix a presentation
G = hx1 , x2 , . . . , xu | r1 , r2 , . . . , rv i.
Since ϕ is surjective, then G′ is generated by ϕ(x1 ), . . . , ϕ(xu ). Namely, G′ can bepresented as
G′ = hϕ(x1 ), ϕ(x2 ), . . . , ϕ(xu ) | s1 , s2 , . . . , sv′ i.

TWISTED ALEXANDER POLYNOMIAL

3

For convenience, we also write xi for ϕ(xi ), that is, we consider that G′ is generated
by x1 , . . . , xu . Here each relator ri is written as
Y
εi
ri =
uk sli k u−1
i = 1, 2, . . . , v, 1 ≤ lik ≤ v ′ , uk ∈ Fu , εik = ±1.
k ,
k

k

∂sk
∂
By applying the Fox derivation ∂x
and collecting terms of ∂x, we get
j
j

(3.1)

ϕ



∂ri
∂xj



′

=

v
X

Ai,k

k=1

∂sk
.
∂xj

Here Ai,k (1 ≤ i ≤ v) is a sum of some ε• ϕ(u• ), which does not depend on j. Let
MG and MG′ be the Alexander matrices removed the u-th column:



 

∂r1
∂r1
ρ̃ ⊗ α̃ ∂x
·
·
·
ρ̃
⊗
α̃
∂xu−1
1




..
..
..
MG = 

.
.


.
 
∂rv
v
ρ̃ ⊗ α̃ ∂x· · · ρ̃ ⊗ α̃ ∂x∂ru−1
1


∂s1
ρ̃′ ⊗ α̃′ ∂x1..
M G′ = .′′ ∂sv′ρ̃ ⊗ α̃ ∂x1


···
..
.

ρ̃′ ⊗ α̃′···ρ̃′ ⊗ α̃′By (3.1), we have


..
.

∂s1
∂xu−1

∂sv′
∂xu−1

 



.
 

MG = AMG′′′

where A = (ρ (Ai,k )) is a nv × nv matrix. For I = (i1 , i2 , . . . , in(u−1) ), as is easily
shown, X
IK
det MG= det AI MG′ =± det AIK det MG′
K

where K = (k1 , k2 , . . . , kn(u−1) ) and AIK is the matrix consisting of the k1 , k2 , . . . ,Ikn(u−1) -th columns of AI . It follows that if det MG′ has a common divisor P forI
all I, then so does det MG . Moreover, the denominator of ∆G,ρ is equal to that of∆G′ ,ρ′ . This completes the proof.
The corresponding fact about the Alexander polynomial is well known. Let
G(K) be the knot group π1 (S 3 − K) of a knot K in S 3 . For any knots K, K ′ , ifthere exists a surjective homomorphism from G(K) to G(K ′ ), then the Alexanderpolynomial of K ′ is divisible by that of K. Murasugi mentions that if there exists a
surjective homomorphism from a knot group G(K) to the trefoil knot group, then
the twisted Alexander polynomial of G(K) is divisible by that of the trefoil knot
group. The main theorem is a generalization of the above.
We remark geometric settings of a surjective homomorphism between groups.
Here we mention degree 1 maps. Let X and Y be d-dimensional compact manifolds.
Suppose that f : X → Y is a degree 1 map. It is easy to see that its induced
homomorphism f∗ : π1 (X) → π1 (Y ) is a surjective homomorphism.
In the knot group case, there exist the following situations except for degree
1 maps. First, there exists a surjective homomorphism from any knot group to
the trivial knot group which is the infinite cyclic group. Secondly, if a knot K
is a connected sum of K1 and K2 , then its knot group G(K) is an amalgamated4

TERUAKI KITANO, MASAAKI SUZUKI AND MASAAKI WADA

product of G(K1 ) and G(K2 ). Then there exists a surjection from G(K) to eachfactor group. Thirdly, if a knot K is a periodic knot of order n, then there exists
a surjective homomorphism from G(K) to G(K∗ ) where K∗ is its quotient knot ofK.
4. Another proof from the view point of Reidemeister torsion
In this section, we prove our theorem in the knot group case. It is done by using
the Mayer-Vietoris argument of Reidemeister torsion.
Here let us consider a knot K in S 3 and its exterior E(K). For the knot group
G(K) = π1 E(K), we choose and fix a Wirtinger presentation
G(K) = hx1 , . . . , xu | r1 , . . . , ru−1 i.
The abelianization homomorphism
αK : G(K) → H1 (E(K), Z) ∼= Z = hti
is given by αK (x1 ) = · · · = αK (xu ) = t. If we have no confusion, we write simply
α for αK as previous section. In this section, we take a unimodular representation
ρ : G(K) → SL(n; F) over a field F. As the definition of the twisted Alexanderpolynomial, we consider the tensor representationρ ⊗ α : G → GL(n; F[t, t−1 ]) ⊂ GL(n; F(t)).
Here F(t) denotes the rational function field over F. If ρ ⊗ α is an acyclic representation over F(t), that is, all homology groups over F(t) of E(K) twisted byρ ⊗ α are vanishing, then Reidemeister torsion of E(K) for ρ ⊗ α can be defined.
Furthermore the following equality holds. See [3, 5] for more details of definitions
and proofs.
Theorem 4.1. If ρ ⊗ α is an acyclic representation, then we haveτρ⊗α (E(K)) = ∆G(K),ρ (t)up to a factor ±tnk(k ∈ Z) if n is odd and up to only tnk if n is even.

From this theorem, we prove the main theorem as divisibility of Reidemeister
torsion in the knot group case. Here we take a surjective homomorphism ϕ :G(K) → G(K ′ ). By changing an orientation of meridians if we need, we may
assume that αK ′ ◦ ϕ = αK . Let ρ′ : G(K ′ ) → SL(n; F) be a representation. For
simplicity, we write a representation ρ for its composition ρ′ ◦ ϕ.
Now we consider 2-dimensional CW-complexes X(K) and X(K ′ ) defined by their
Wirtinger presentations. It is well-known that these complexes are simple homotopy equivalent to the knot exteriors. Then these Reidemeister torsions of X(K)and X(K ′ ) are equal to the twisted Alexander polynomials respectively. Here we
consider twisted homologies of these complexes by using their CW-complex structure. The coefficient V is a 2n-dimensional vector space over a rational function
field F(t). When V is regarded as a G(K)-module by using ρ, it is denoted by Vρ .
The homomorphism ϕ induces a chain map ϕ∗ : C∗ (X(K), Vρ ) → C∗ (X(K ′ ), Vρ′ ).
We take a tensor representation ρ⊗αK : G(K) → GL(n; F(t)). Assume that ρ⊗αKand ρ′ ⊗ αK ′ are acyclic representations. Then we can prove the following.
Theorem 4.2. τ (X(K); Vρ⊗αK )/τ (X(K ′ ); Vρ′ ⊗αK ′ ) is a polynomial in F[t, t−1 ].
We show the following proposition first.

TWISTED ALEXANDER POLYNOMIAL

5

Proposition 4.3. The chain map ϕ∗ : C∗ (X(K), Vρ⊗αK ) → C∗ (X(K ′ ), Vρ′ ⊗αK ′ )is surjective.
Proof. It is clear that ϕ induces an isomorphism on the 0-chains, and a surjection
on the 1-chains. Then we only need to prove this proposition on the 2-chains.
We take a non-trivial 2-chain z ∈ C2 (X(K ′ ), Vρ′ ⊗αK ′ ). By the acyclicity of thechain complex C∗ (X(K ′ ), Vρ′ ⊗αK ′ ), the boundary map ∂ : C2 (X(K ′ ), Vρ′ ⊗αK ′ ) →C1 (X(K ′ ), Vρ′ ⊗αK ′ ) is injective. Then the image ∂z is non-trivial in C1 . On theother hand, by the surjectivity of
ϕ : C1 (X(K), Vρ⊗αK ) → C1 (X(K ′ ), Vρ′ ⊗αK ′ ),there exists a 2-chain w ∈ C2 (X(K), Vρ⊗αK ) such that ϕ∗ (w) = z. By the commutativity of maps, in C2
ϕ∗ (∂w) = ∂ϕ∗ (w) = ∂∂z = 0.
Then we have ∂w = 0. Hence by the acyclicity, there exists w̃ ∈ C∗ (X(K), Vρ⊗αK )such that ∂ w̃ = w. Again by the commutativity, ϕ(w̃) = z. Therefore ϕ∗ issurjective.
Proof of Theorem 4.2. From the above proposition, we can take the kernel D∗of this chain map ϕ∗ and obtain a short exact sequence
0 → D∗ → C∗ (X(K), Vρ⊗αK ) → C∗ (X(K ′ ), Vρ′ ⊗αK ′ ) → 0.
Here we recall the following fact. For a short exact sequence 0 → C∗′ → C∗ → C∗′′of finite chain complexes, if two of them are acyclic complexes, then the third one
is also acyclic. Furthermore, among the torsion of them, it holds that
τ (C∗ ) = τ (C∗′ )τ (C∗′′ )up to some factor.
By applying the property of the product of torsion, we have
τ (X(K); Vρ⊗αK ) = τ (X(K ′ ); Vρ′ ⊗αK ′ )τ (D; Vρ⊗αK ).
We only need to prove that τ (D; Vρ⊗αK ) is a polynomial. By the definition,
0-chains D0 is vanishing. Becauseϕ∗ : C0 (X(K), Vρ⊗αK ) → C0 (X(K ′ ), Vρ′ ⊗αK ′ )is isomorphism. Hence by definition, its torsion is the determinant of D2 → D1 .
Therefore it is a polynomial.
Remark 4.4. By the similar argument, we can prove that if ϕ : G(K) → G(K ′ ) is aninjective homomorphism, then τ (X(K ′ ); Vρ⊗αK ′ )/τ (X(K); Vρ⊗αK ) is a polynomial.
5. Examples
In this section, we show some examples of the twisted Alexander polynomials
and an application for Theorem 3.1. We consider the problem: is there a surjective
homomorphism from G(K) to G(K ′ ) for two given knots K, K ′ ? Here we study the
problem in case when K ′ is the figure eight knot 41 . It is investigated by Murasugi
when K ′ is the trefoil knot 31 . The numbering of the knots follows that of Rolfsen’sbook [7].
If the classical Alexander polynomial of K can not be divided by that of K ′ , wecan determine whether or not there exists a surjective homomorphism from G(K) toG(K ′ ). In the knot table in [7], up to 9 crossings, the classical Alexander polynomial

6

TERUAKI KITANO, MASAAKI SUZUKI AND MASAAKI WADA

of any knot is not divisible by that of G(41 ) except for 818 , 821 , 912 , 924 , 937 , 939 and940 . That is to say, except for 818 , 821 , 912 , 924 , 937 , 939 and 940 , there exists nosurjective homomorphism from such a knot group to G(41 ).
Next, we consider a representation ρ : G(K) → SL(2; Z/pZ) and the twisted
Alexander polynomial associated to ρ. Theorem 3.1 says that if the numerator of∆G(K),ρ for all representations ρ : G(K) → SL(2; Z/pZ) fixed some prime p can not
be divided by the numerator of ∆G(K ′ ),ρ′ for a certain representation ρ′ : G(K ′ ) →SL(2; Z/pZ), then there exists no surjective homomorphism from G(K) to G(K ′ ).
Let us compute the twisted Alexander polynomials ∆G(41 ),ρ′ for a certain representation ρ′ : G(41 ) → SL(2; Z/7Z). The knot group G(41 ) admits a presentation
−1
−1 −1
−1 −1
G(41 ) = hx1 , x2 , x3 , x4 | x4 x2 x−1
4 x1 , x1 x2 x1 x3 , x2 x4 x2 x3 i.

We can check easily that the following is a representation of G(41 ):








1 1
1 0
4 4
2 4
′
′
′
′
ρ (x1 ) =, ρ (x2 ) =, ρ (x3 ) =, ρ (x4 ) =
.
0 1
3 1
3 5
5 0
Then we obtain the Alexander matrix

6
0
2t

0
6
5t

 3t + 1
3t
t
M =
 4t
2t
+
1
0


0
0
3t + 1
0
0
4t

4t
0
t
t
3t
2t + 1

0
0
6
0
6
0

0 6t + 1
0
0
0
0
6
0
0
t
6
3t


6t
6t + 1 


0
.

0


0
t

The numerator P of the twisted Alexander polynomial ∆G(41 ),ρ′ is the determinantof M4 obtained from M by removing the last two columns. Then we get
P = t4 + t3 + 3t2 + t + 1.
Moreover, we calculate the numerator of the twisted Alexander polynomials ofG(821 ) for all representations G(821 ) → SL(2; Z/7Z) and get 24 polynomials. These
calculations are made by author’s computer program and the same results are obtained by Kodama Knot program [4]. None of them can be divided by P , thenwe conclude that there exists no surjective homomorphism from G(821 ) to G(41 ).
Similarly, if we deal with p = 5, 7, we get there exists no surjective homomorphismfrom G(912 ), G(924 ), G(939 ) to G(41 ). On the other hand, we can give explicitly surjective homomorphisms from the rest knot groups G(818 ), G(937 ), G(940 ) to G(41 ).
Then we can determine whether or not there exists a surjective homomorphismfrom the group of all knot with up to 9 crossings to G(41 ).
Acknowledgements. The authors would like to express their thanks to Prof.
Sadayoshi Kojima and Prof. Dieter Kotschick for their useful comments.
References
1. R. Crowell and R. Fox, Introduction to knot theory, GTM 57, Springer
2. H. Goda, T. Kitano and T. Morifuji, Reidemeister torsion, Twisted Alexander polynomialand fibered knots, to appear in Comment. Math. Helv.
3. P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and CassonGordon invariants, Topology 38 (1999), 635–661.
4. http://www.math.kobe-u.ac.jp/HOME/kodama/knot.html
5. T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174(1996), 431–442
6. X.S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin.
(Engl. Ser.) 17 (2001), 361–380.

TWISTED ALEXANDER POLYNOMIAL

7

7. D. Rolfsen, Knots and links, Publish or Perish, Inc.
8. M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994),
241–256.
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-43 Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan
E-mail address: kitano@is.titech.ac.jp
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba,
Meguro-ku, Tokyo, 153-8914 Japan
E-mail address: macky@ms.u-tokyo.ac.jp
Department of Information and Computer Sciences, Nara Women’s University, KitaUoya Nishimachi, Nara 630-8506, Japan
E-mail address: wada@ics.nara-wu.ac.jp

Preprint Series, Graduate School of Mathematical Sciences, The University of Tokyo
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2004–25 Taro Asuke: The Godbillon-Vey class of transversely holomorphic foliations.
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problem for a hyperbolic equation with variable coeﬃcients.
2004–28 Naoki Heya: The absolute continuity of a measure induced by inﬁnite dimensional stochastic diﬀerential equations.
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2004–30 Yasuharu Nakae: Taut foliations of torus knot complements.
2004–31 Teruhisa Tsuda: Universal characters and q-Painlevé systems.
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2004–36 Teruaki Kitano, Masaaki Suzuki and Masaaki Wada: Twisted Alexander polynomial and surjectivity of a group homomorphism.

The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the former
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