arXiv:1602.05884v1 [math.GT] 18 Feb 2016

AN EXPLICIT RELATION BETWEEN KNOT GROUPS IN
LENS SPACES AND THOSE IN S 3
YUTA NOZAKI

Abstract. We consider a cyclic covering map (Σ, K) → (Σ′ , K ′ ) between pairs of a 3-manifold and a knot, and describe the fundamental
group π1 (Σ \ K) in terms of π1 (Σ′ \ K ′ ). As a consequence, we give
an alternative proof for the fact that a certain knot in S 3 cannot be
represented as the preimage of any knot in a lens space. In our proofs,
the subgroup of a group G generated by the commutators and the pth
power of each element of G plays a key role.

Contents
1. Introduction
2. Lemmas in group theory and knot theory
3. Main theorem and corollaries
Appendix A. Symmetric groups and related contents
Appendix B. Taking Cp repeatedly
References

1
3
5
8
9
10

1. Introduction
Let K be a torus knot and p ≥ 2 be an integer. Hartley [7] created a
decision argument to know whether K can be represented as the preimage of
a knot in the lens space L(p, q) for some q. For example, the trefoil T3,2 does
not appear as the preimage of any knot in L(2, 1) ∼
= RP 3 . The Alexander
polynomial of a torus knot was used in his proof.
Let K be a knot such that the outer automorphism group Out(π1 (S 3 \K))
is trivial. For instance, 932 , 933 and 24 more prime knots with 10 crossings
(and their mirror images) satisfy this condition (see Kawauchi [11, Appendix F.2] or Kodama-Sakuma [12, Table 3.1]). Then it follows from Borel’s
theorem (see Conner-Raymond [6, Theorem 3.2]) that K cannot be represented as the preimage of any knot in any lens space (see [11, Theorems 10.6.2 and Theorem 10.6.6 (1)]).
The purpose of this paper is to deduce the above facts from a single
result. Let G be a group and p ≥ 1 be an integer. (p is not necessarily
2010 Mathematics Subject Classification. 57M25, 57M27.
Key words and phrases. Knot group, lens space, covering space, pth power.
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Y. NOZAKI

prime.) We introduce the subgroup Cp (G) generated by the commutators
and the pth power of each element of G. The case of p = 2 was studied, for
example, in Sun [19] and Haugh-MacHale [9], from a purely algebraic point
of view. Moreover, the case where p is prime can be found in Stallings [18]
and Cochran-Harvey [4] (see Remark 2.2 and Appendix B). We apply Cp to
the fundamental group of the complement of a knot in a 3-manifold. Let
π : Σ → Σ′ be a p-fold cyclic covering, where Σ is an integral homology
3-sphere. The main result of this paper is the following.
Theorem 1 (Theorem 3.1). Let K ′ be a knot in Σ′ with the connected
preimage K := π −1 (K ′ ). Then the image of π∗ : π1 (Σ \ K) ֌ π1 (Σ′ \ K ′ )
coincides with Cp (π1 (Σ′ \ K ′ )).
Considering the case of Σ = S 3 , Σ′ = L(p, q), we obtain the following
corollaries as mentioned before.
Corollary 1 (Corollary 3.4). Let m, n, p ∈ Z≥2 with gcd(m, n) = 1. There
exists an integer q and a knot K ′ in L(p, q) such that π −1 (K ′ ) is isotopic to
the torus knot Tm,n or its mirror image if and only if gcd(mn, p) = 1.
Furthermore, Chbili [3] determined the above q, which also follows from
Corollary 1 and some other results (see Proposition 3.6 for details).
Corollary 2 (Corollary 3.3). A knot K in S 3 with Out(π1 (S 3 \ K)) = 1
cannot be represented as the preimage of any knot in any lens space.
Another proof of a special case of Corollary 2 is given in Appendix A.1.
We expect that the use of Theorem 1 in the appendix can be extended to
some other knots.
When a knot K is represented as a preimage, there is still the question
of whether K ′ is uniquely determined. Sakuma [16], Boileau and Flapan [1]
independently proved that if the preimages of oriented knots K0′ , K1′ are
the same prime knot, then there exists a diffeomorphism (L(p, q), K0′ ) →
(L(p, q), K1′ ) preserving the orientations both of L(p, q) and Ki . Note that
K0′ and K1′ are not necessarily ambient isotopic to each other. (They also
showed that the same is true for a composite knot under a condition regarding “slopes”.) Furthermore, Manfredi [13] recently constructed two knots in
a lens space such that they are not ambient isotopic to each other but their
preimages are the unknot.
Our work is closely related to freely periodic actions on (S 3 , K). Indeed,
such an action of period p induces a cyclic covering (S 3 , K) → (L(p, q), K ′ )
for some q. Hartley [7] gave a list of possible free periods of prime knots up
to 10 crossings.
Acknowledgments. The author would like to express his gratitude to
Takuya Sakasai for his various suggestions. Also, he would like to thank
Makoto Sakuma for pointing out some mistakes in his first draft and answering his questions about the symmetry of knots. Finally, this work was
supported by the Program for Leading Graduate Schools, MEXT, Japan.

KNOT GROUPS IN LENS SPACES AND THOSE IN S 3

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2. Lemmas in group theory and knot theory
First of all, we set up notation and prepare basic lemmas used in Section 3.
2.1. Notation. Let X be a topological space and G be a group. We set
H∗ (X) := H∗ (X; Z) and H∗ (G) := H∗ (G; Z) with the trivial action G y
Z. For a knot K in S 3 , let G(K) denote the fundamental group of the
complement of K, that is, G(K) = π1 (S 3 \ K). (We omit a base point if we
need not to care it.) We denote by Zn the cyclic group of order n.
2.2. Definition of Cp and related lemmas. We first introduce an important subgroup and give some examples.
Definition 2.1. For a group G and p ∈ Z≥1 , let Cp (G) denote the subgroup
of G generated by the set {gp | g ∈ G} ∪ {[g, h] | g, h ∈ G}, where [g, h] :=
ghg −1 h−1 . A group G is called a Cp -group if there is a group G′ such that
G∼
= Cp (G′ ).
Note that C1 (G) = G and C2 (G) = {g2 | g ∈ G} .
Remark 2.2. C2 (G) is denoted by G2 in [19] and by S(G) in [9]. For a
prime p, Cp (G) coincides with the 1st term of the p-lower central series in
[18] and with the 1st term of the derived p-series in [4].
Example 2.3. Let us determine the subgroup Cp Sn of the nth symmetric
group Sn . In the case where p is odd, Cp Sn contains all transpositions, and
thus Cp Sn = Sn . In the case where p is even, we conclude Cp Sn = An
since An = [Sn , Sn ] ≤ Cp Sn ≤ An , where An is the nth alternating group.
We next state two basic lemmas without proofs.
Lemma 2.4. For any group G, the subgroup Cp (G) is characteristic, and
the quotient group G/Cp (G) is abelian. Moreover, if G is finitely generated,
then G/Cp (G) is isomorphic to a finite direct sum of finite cyclic groups
whose orders are divisors of p.
Some simple examples of G/Cn (G) are listed in Table 2.2.
G/Cp (G)
G
Zn
Zgcd(n,p)
⊕{prime
numbers}
(Q>0 , ×) (Zp )
(R>0 , ×)
0
Table 2.1. Examples of G/Cp (G). Note that gcd(0, p) = p.

Lemma 2.5. For groups G and H, the following hold.
(1) Cp (G × H) = Cp (G) × Cp (H).
(2) A homomorphism f : G → H induces the homomorphism Cp (G) →
Cp (H) denoted by Cp (f ). Moreover, if f is surjective (resp. injective), then Cp (f ) is also surjective (resp. injective).

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Y. NOZAKI

The following lemma is a refinement of the well-known fact [14, 13.5.8]
for a complete group H.
Lemma 2.6. Let G, H be groups such that H ⊳ G and Adg |H ∈ Inn(H)
for any g ∈ G. Then the sequence of groups
φ

ψ

1 → Z(H) −
→ H × CG (H) −
→G→1
is exact, where CG (H) is the centralizer of H in G, and φ(h) := (h, h−1 ),
ψ(h, g) := hg.
The next lemma follows from the above lemmas, which is a generalization
of [9, Theorem 1].
Lemma 2.7. Let G, H be as in Lemma 2.6 and suppose Cp (G) = H,
Z(H) = 1. Then Cp (H) = H.
Example 2.8. Let us determine whether Sn is a Cp -group or not. By
Example 2.3, if p is odd or n = 1, 2, then Sn is a Cp -group. In the other
cases except n = 6, Sn is not a Cp -group by Lemma 2.7 and the fact
that Sn is complete for n 6= 2, 6. Furthermore, S6 is not a Cp -group (see
Appendix A.2).
Lemma 2.9. For a group G whose abelianization is isomorphic to Z⊕r , the
quotient group G/Cp (G) is isomorphic to (Zp )⊕r .
Proof. Since the codomain of the projection G ։ G/Cp G isLabelian, it
r
factors through the abelianization Z⊕r of G. Hence, G/Cp G ∼
= i=1 Zpi for
⊕r
some pi | p. The abelianization G ։ Z induces a surjection G/Cp G ։
Z⊕r /Cp (Z⊕r ) = (Zp )⊕r . It follows that pi = p for i = 1, . . . , r.

Remark 2.10. If p is prime, then Lemma 2.9 directly follows from [18,
Theorem 3.4] or [4, Lemma 2.3].
2.3. Links in a rational homology 3-sphere. The following lemma is a
special case of [8, Theorem 2.1].
Lemma 2.11. Let L′ be an r-component link in a rational homology 3incl∗
sphere Σ′ such that H1 (L′ ) −−−→
H1 (Σ′ ) is surjective. Then the following
holds:

Z
if ∗ = 0,



Z⊕r
if ∗ = 1,
H∗ (Σ′ \ L′ ) ∼
=
⊕(r−1)

Z
if ∗ = 2,



0
otherwise.
Remark 2.12. If H1 (Σ′ ) ∼
= Zp and r = 1, then a meridian of a knot L′
∼
corresponds to ±p ∈ Z = H1 (Σ′ \ L′ ).
Remark 2.13. Let Σ′ , L′ be as in Lemma 2.11. Suppose that the universal
cover of Σ′ is S 3 , and the link π −1 (L′ ) in S 3 is non-split. Then H∗ (π1 (Σ′ \L′ ))
is isomorphic to H∗ (Σ′ \ L′ ). Indeed, since the total space S 3 \ π −1 (L′ ) is an
Eilenberg-MacLane space, so the base space Σ′ \ L′ is.

KNOT GROUPS IN LENS SPACES AND THOSE IN S 3

5

3. Main theorem and corollaries
Recall that we are interested in knowing whether a knot K in S 3 is represented as the preimage of a knot K ′ in a lens space.
3.1. Application of Cp to covering spaces. Let π : Σ → Σ′ be a p-fold
cyclic covering, where Σ is an integral homology 3-sphere. One can check
that H∗ (Σ′ ) ∼
= H∗ (L(p, q)).
Theorem 3.1. Let K ′ be a knot in Σ′ with the connected preimage K :=
π −1 (K ′ ). Then the image of π∗ : π1 (Σ \ K) ֌ π1 (Σ′ \ K ′ ) coincides with
Cp (π1 (Σ′ \ K ′ )).
Proof. Set G := π1 (Σ \ K) and G′ := π1 (Σ′ \ K ′ ). Since there is a surjection
(G′ )ab ։ G′ /Cp G′ , by Lemma 2.11, G′ /Cp G′ ∼
= Zp′ for some p′ | p.
On the other hand, the covering map π induces an exact sequence
π

∗
1 → G −→
G′ → Zp → 1.

Using Cp G ≤ π∗ (G) ≤ G′ , we have
p′ = [G′ : Cp G′ ] = [G′ : π∗ (G)][π∗ (G) : Cp G′ ] ≥ p.
It follows that p′ = p, and thus π∗ (G) = Cp G′ .


π −1 (K ′ )

Remark 3.2. Let r be the number of connected components of
and fix the isomorphism H1 (Σ′ ) ∼
= Zp so that [K ′ ] corresponds to k ′ , where
k′ := p/ord([K ′ ]) ∈ Z. Then we have r = k′ . In particular, π −1 (K ′ )
is connected if and only if [K ′ ] generates H1 (Σ). Indeed, we first obtain
the canonical isomorphism H1 (Σ′ ) ∼
= π1 (Σ′ )/π∗ π1 (Σ) since the projection
′
′
π1 (Σ ) ։ π1 (Σ )/π∗ π1 (Σ) ∼
= Zp factors through π1 (Σ′ )ab . Choose a connected component K of π −1 (K ′ ) and a base point ∗ ∈ K, and set ∗′ := π(∗).
(The other connected components are τ r (K), τ 2r (K), . . . , τ p−r (K), where τ
is a generator of the deck transformation group.) It follows from π(K) =
K ′ p/r as based loops that pr × k′ = 0, and thus r | k′ . On the other hand,
′
the equality kp′ × [K ′ ] = 0 implies that [K ′ ]p/k ∈ π1 (Σ′ , ∗′ ) belongs to the
′
image of π∗ . Hence, K ′ p/k can be lifted to Σ. Then we conclude pr | kp′ , and
thus k′ | r.
The case of Σ = S 3 is discussed in [13, Proposition 2.2].
As mentioned in Section 1, the following corollary is already known. However, using Theorem 3.1, we give an alternative proof of it.
Corollary 3.3. A knot K in S 3 with Out(G(K)) = 1 is not represented as
the preimage of any knot in any lens space.
Proof. Since Out(G(Tm,n )) ∼
= Z2 is proven in [17], K is not a torus knot,
and thus Z(G(K)) = 1 by [2]. Hence, G(K) is complete.
Assume that there exists a knot K ′ in L(p, q) whose preimage is isotopic
to K, and set G′ := π1 (L(p, q) \ K ′ ). Then we have G(K) = Cp G′ by
Theorem 3.1. Since G(K) is complete, by Lemme 2.7, we conclude Cp G′ =
G′ . However, this contradicts Lemmas 2.11 and 2.9.


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Y. NOZAKI

3.2. Application of Theorem 3.1 to torus knots. The aim of this subsection is to prove the following corollary.
Corollary 3.4 ([7, Theorem 3.1]). Let m, n, p ∈ Z≥2 with gcd(m, n) = 1.
There exists an integer q and a knot K ′ in L(p, q) such that π −1 (K ′ ) is isotopic to the torus knot Tm,n or its mirror image if and only if gcd(mn, p) = 1.
We have to prepare the next lemma in homology of groups.
Lemma 3.5. Let m, n, p be as in Corollary 3.4 and G be a group admitting
the following exact sequence
(3.1)

1 → Zm ∗ Zn ֒→ G → Zp → 1.

If gcd(mn, p) = 1, then H∗ (G) ∼
= H∗ (Zmnp ). Moreover, the converse is true
under the additional condition Cp (G) = Zm ∗ Zn .
Proof. Let g ∈ G and we want to show that Adg |Zm ∗Zn ∈ Inn(Zm ∗ Zn ). It
is known that Aut(Zm ∗ Zn ) is generated by inner automorphisms and φi,j
for i ∈ (Zm )× , j ∈ (Zn )× where φi,j is defined by φi,j (a) = ai , φi,j (b) = bj .
Therefore, Adg |Zm ∗Zn = Adw ◦ φi,j for some 1 ≤ i < m, 1 ≤ j < n and
w ∈ Zm ∗ Zn , and then the induced automorphism (Adg |Zm ∗Zn )ab sends n, m
to in, jm respectively. On the other hand, (Adg )ab = idZmnp .
Let us consider the five-term exact sequence
0 = H2 (Zp ) → (Zmn )Zp → Zmnp → Zp → 0
for the short exact sequence (3.1). Comparing the order of each group, we
conclude that (Zmn )Zp = Zmn . It follows that in = n and jm = m in Zmn ,
and thus i = j = 1. Hence, the action G y H∗ (Zm ∗ Zn ) is trivial, and so is
Zp y H∗ (Zm ∗ Zn ).
The condition gcd(mn, p) = 1 implies that the E 2 -term of the LyndonHochschild-Serre spectral sequence for (3.1) is as follows:

Z
if s = t = 0,



Z
mn if s = 0, t ≥ 1 is odd,
2
Es,t
= Hs (Zp ; Ht (Zm ∗ Zn )) =
Z p
if s ≥ 1 is odd, t = 0,



0
otherwise,
and then all differentials dr are zero for r ≥ 2. Using gcd(mn, p) = 1 again,
we have H∗ (G) ∼
= H∗ (Zmnp ).
We next prove the latter of the above statement. Suppose Cp (G) =
Zm ∗Zn . Since we know that Adg |Zm ∗Zn ∈ Inn(Zm ∗Zn ) and Z(Zm ∗Zn ) = 1,
by Lemma 2.7, we have Cp (Zm ∗ Zn ) = Zm ∗ Zn . Finally, the abelianization
induces a surjection Zm ∗ Zn /Cp (Zm ∗ Zn ) ։ Zmn /Cp Zmn = Zgcd(mn,p) , and
thus gcd(mn, p) = 1.


Note that the following proof does not assert that if gcd(mn, p) 6= 1, then
G(Tm,n ) is not a Cp -group.

KNOT GROUPS IN LENS SPACES AND THOSE IN S 3

7

Proof of Corollary 3.4. If gcd(mn, p) = 1, then a construction of a desired
knot K ′ is given in [7, Theorem 3.1].
Suppose there exists K ′ as in the statement, and set π1′ := π1 (L(p, q)\K ′ ).
The covering map π : S 3 \ Tm,n → L(p, q) \ K ′ induces the exact sequence
π

∗
1 → G(Tm,n ) = ha, b | am = bn i −→
π1′ → Zp → 1.

Since the center Z(G(Tm,n )) = ham i = Z is characteristic in G(Tm,n ), the
subgroup N := π∗ (ham i) of π1′ is normal. We deduce the exact sequence
π

∗
1 → ha, b | am = 1 = bn i −→
π1′ /N → Zp → 1

from the 3rd isomorphism theorem. The group G := π1′ /N admits (3.1) and,
by Theorem 3.1, it satisfies
Cp (G) = Cp (π1′ )/N = G(Tm,n )/ ham i = Zm ∗ Zn .
Hence, by Lemma 3.5, it suffices to prove H∗ (G) ∼
= H∗ (Zmnp ).
By Lemma 2.11 and Remark 2.13, the five-term exact sequence for
(3.2)
is as follows:

1 → N → π1′ → G → 1
φ

→ Z → H1 (G) → 0.
0 → H2 (G) → ZG −

Qn
−1 m
Since the generator am of N is decomposed as
in π1′ for a
i=1 gi µgi
′
′
meridian µ of K and some gi ∈ π1 , we conclude φ(1) = mnp by Remark 2.12.
Therefore, ZG = Z (namely, G y H1 (N ) is trivial), H2 (G) = 0 and H1 (G) =
Zmnp . The E 2 -term of the Lyndon-Hochschild-Serre spectral sequence for
(3.2) is as follows:


Hs (G; H0 (N )) if t = 0,
2
Es,t
= Hs (G; Ht (N )) = Hs (G)
if t = 1,


0
otherwise.

Moreover, the differential d2s,0 is an isomorphism for s ≥ 3, and Hs (G; H0 (N )) =
H0 (N )G is isomorphic to Z. Therefore, the action G y H0 (N ) is trivial,
and thus Hs (G; H0 (N )) = Hs (G). We inductively conclude that H1 (G) ∼
=
H3 (G) ∼

= · · · and H2 (G) ∼
= H4 (G) ∼
= ···.
We finally discuss an integer q in Corollary 3.4.
Proposition 3.6 ([3, Remark 3.7]). Let |m|, |n|, p ∈ Z≥2 with gcd(m, n) =
1. Suppose that the torus knot Tm,n is isotopic to the preimage of a knot in
L(p, q) for some q. Then q ≡ m(1 − pp∗ )/n mod p, where p∗ is an integer
satisfying pp∗ ≡ 1 mod n.

Proof. We first remark that gcd(mn, p) = 1 by Corollary 3.4. Hence, p∗
exists and gcd(p, m(1 − pp∗ )/n) = gcd(p, 1 − pp∗ ) = 1 holds.
Since Tm,n is a prime knot, by [16, Theorem 5] or [1, Theorem 1], q is
uniquely determined up to taking its inverse in Zp . (Recall that L(p, q0 ) ∼
=
L(p, q1 ) as oriented manifolds if and only if q0 ≡ q1±1 mod p.) Therefore, it

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Y. NOZAKI

suffices to see that there exists a knot in L(p, q) whose preimage is Tm,n for
q = m(1 − pp∗ )/n. Let q := m(1 − pp∗ )/n. We now conclude p | m − nq.
Hence, using a suitable n-braid (see [13, Proposition 3.3]), we construct a
desired knot in L(p, q).

Appendix A. Symmetric groups and related contents
A.1. Another proof of Corollary 3.4 for the trefoil. The purpose of
this subsection is to introduce another use of Theorem 3.1. We first remark
that a quotient group of a Cp -group G is not a Cp -group in general. However,
if H ⊳ G is characteristic, then G/H is also a Cp -group, that is, the following
lemma holds.
Lemma A.1 (see [19, Theorem 1]). Let G be a Cp -group and G ։ G′ be
a surjective homomorphism whose kernel is a characteristic subgroup of G.
Then G′ is also a Cp -group.
Theorem A.2. The trefoil T3,2 cannot be represented as the preimage of a
knot in L(p, q) for any p ∈ 2Z≥1 , q ∈ Z.
Proof. Assume that there exists a knot K ′ in L(p, q) whose preimage is
isotopic to T3,2 . By Theorem 3.1, G(T3,2 ) = a, b | a3 = b2 is a Cp -group.
We define a surjective homomorphism φ : G(T3,2 ) ։ S3 by φ(a) = (123),
φ(b) = (12). One can check that Ker φ is characteristic. Indeed, by [17], it
suffices to show that φ(θ(w)) = 1 for any w ∈ Ker φ, where θ ∈ Aut(G(T3,2 ))
is defined by θ(a) = a−1 , θ(b) = b−1 . Thus, S3 is also a Cp -group by
Lemma A.1. However, this contradicts Example 2.8.

A.2. The 6th symmetric group. In this subsection, we prove that S6
is not a Cp -group for even p (see Example 2.3). The following argument is
based on [9, Theorem 3]. Let α denote the homomorphism A6 ֒→ S6 ֌
Aut(S6 ).
Lemma A.3 (see [9, p. 125]). Cp (Aut(S6 )) = Im α for p ∈ 2Z≥1 .
Proof. It follows from [S6 , S6 ] = A6 that Cp (Aut(S6 )) ≥ Im α. One has to
show that φp and [φ, φ′ ] belong to Im α for any φ, φ′ ∈ Aut(S6 ). We prove
only φp ∈ Im α for φ ∈
/ Inn(S6 ). (The remainder is shown similarly.)
It is known that there exists ψ ∈ Aut(S6 ) \ Inn(S6 ) whose order is 10.
Since Out(S6 ) ∼
= Z2 , we have ψ 2 = Adτ for some τ ∈ S6 . It follows from
10
ψ = idS6 that τ 5 = 1, and thus τ ∈ A6 . Hence, φ is written as φ = ψ ◦Adσ
for some σ ∈ S6 . Then we have
(ψ ◦ Adσ )p = (ψ 2 ◦ Adψ−1 (σ)σ )p/2 = Ad(τ ψ−1 (σ)σ)p/2 .
Since ψ −1 (σ)σ ∈ A6 , we conclude φp ∈ Im α.
Lemma A.4 (see [9, Theorem 3]). For any
Inn(G), where p ∈ Z≥1 .
Proposition A.5. S6 is not a Cp -group.


Cp -group

G,

Cp (Aut(G))

≥

KNOT GROUPS IN LENS SPACES AND THOSE IN S 3

9

Proof. Assume S6 is a Cp -group. By Lemma A.4, we have Cp (Aut(G)) ≥
Inn(G), and thus
2 = [Aut(S6 ) : Inn(S6 )] = [Aut(S6 ) : Cp Aut(S6 )][Cp Aut(S6 ) : Inn(S6 )].
On the other hand, we have Aut(S6 )/Cp Aut(S6 ) ։ Z2 /Cp Z2 = Z2 . Hence,
Cp Aut(S6 ) = Inn(S6 ) ∼

= S6 . This contradicts Lemma A.3.
A.3. Alternating groups. We finally determine the group Cp (An ).
Proposition A.6. For p ∈ Z≥2 and n ∈ Z≥3 ,


if n = 3 and 3 | p,
1
p
C (An ) = V4 if n = 4 and 3 | p,


An otherwise,

where V4 is the Klein four-group.

Proof. The case of n = 3 is obvious since A3 = Z3 . In the case of n ≥ 5,
p
it is well known that Aab
n = 0, and thus C An = An . Since [A4 , A4 ] = V4 ,
p
p
C A4 is isomorphic to V4 or A4 . Here, C A4 contains cyclic permutations of
order three if and only if 3 ∤ p.

Appendix B. Taking Cp repeatedly
In this section, we discuss the derived p-series of a knot group. In [4], for
a prime p, the derived p-series {G(n) }n≥0 of a group G is defined by
G(0) := G,

G(n+1) := Cp (G(n) ).

We use this notation when p is not only a prime but also a positive integer.
Here, determining
the successive quotients G(n) /G(n+1) and the intersection
T
G(ω) := n≥0 G(n) is a fundamental problem.
Proposition B.1. Let G be a group with Gab ∼
= Z⊕r . Then the abelian
(n)
(n+1)
⊕r
group G /G
surjects onto Zp .

Proof. The abelianization of G induces the homomorphism G(n) ։ (Z⊕r )(n)
for each n. Hence, we obtain G(n) /G(n+1) ։ (Z⊕r )(n) /(Z⊕r )(n+1) ∼
= Z⊕r
p .

Remark B.2. In contrast, the stability of the (usual) derived series of G(K)
depends on a knot K ([5, Corollary 4.8], [15, Proposition 2.7]).
Proposition B.3. Let p be a prime, G be a group with H1 (G) ∼
= Z⊕r and
(ω)
H2 (G) = 0. Then G
coincides with the commutator subgroup of G.
Proof. Since the abelianization α : G ։ Z⊕r induces an isomorphism on
H1 (−; Zp ) and a surjection on H2 (−; Zp ), by [4, Corollary 4.3], α induces an
isomorphism G/G(ω) → Z⊕r /(Z⊕r )(ω) = Z⊕r . Therefore, G(K)(ω) coincides
with Ker α = [G(K), G(K)].


10

Y. NOZAKI

Corollary B.4. For p and G as in Proposition B.3, G(n) /G(n+1) is isomorphic to Z⊕r
p .
Proof. We first have homomorphisms
G(n) /Cp (G(n) ) = G(n) /G(n+1) ֒→ G/G(n+1) և G/G(ω) ∼
= Z⊕r .
′
It follows that G(n) /G(n+1) ∼
= Zp⊕r for some r ′ ≤ r. By Proposition B.1, we
conclude r ′ = r.


Remark B.5. Even if p is not prime, by considering a prime factor of p,
we have G(ω) ≤ [G, G]. On the other hand, if 6 | p, then the equality fails
(1)
(2)
for G = G(31 ), where 31 is the trefoil. Indeed, since S3 = A3 = 1, the
homomorphism φ : G(31 ) ։ S3 defined in the proof of Theorem A.2 induces
G(31 )/G(31 )(ω) ։ S3 . Since S3 is not abelian, G(31 )(ω) is strictly smaller
than the commutator subgroup of G(31 ).
The same argument shows that if G(K) surjects onto G(31 ), then the
equality fails as well. Such knots are found, for example, in [10, Theorem 1.2].
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11

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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: nozaki@ms.u-tokyo.ac.jp

