101

数理解析研究所講究録
第2004巻 2016年 101-107

An explicit relation between knot groups in lens spaces andthose in S^{3}

Yuta Nozaki

Graduate School of Mathematical Sciences, the University of Tokyo

1

Introduction
We consider the symmetry of knots, precisely, free periods. The details of proofs in thisarticle can be found in authors preprint [12]. A knot K in S^{3} is said to have free period

p\in \mathbb{Z}_{\geq 1} if there exists f\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^{3}, K) such that f^{i} has no fixed point for 0<i<p andf^{p}=\mathrm{i}\mathrm{d}_{S^{3} namely, (S^{3}, K) admits a free action by \mathbb{Z}_{p}=\mathbb{Z}/p\mathbb{Z} Whether a knot K has,.

free period p or not is interesting problem and studied by many people.
We first review previous researches on the existence of free periods.

In[7], Hartley

proved that the torus knot T_{m,n} has free period p if and only if \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1.

For

example, the trefoil T_{3,2} does not admit free involution. The Alexander polynomial of atorus knot was used in his proof.

Let K be$\pi$_{1}(S^{3}\backslash K)aknot such that the outerautomorphism group Out (G(K)) of G(K)=is trivial. For instance, 9_{32}, 9_{33} and 24 more prime knots with 10 crossings (andtheir mirror images) satisfy this condition (see Kawauchi [9, Appendix F.2] or Kodama‐

Sakuma [10, Table 3.1]).

Then it follows from Conner‐Raymond[6, Theorem 3.2] andBurde‐Zieschang [3] that K has no free period.
The purpose of this article is to deduce the above facts from a single result.

stating our results,
Sakuma[14],

weprevious researches on the uniqueness of free periods.

review

Boileau andBeforeFlapan [2] independently proved that for an oriented prime

knot K , if f, g\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^{3}, K) have free period p then f is conjugate to g in the subgroup,of \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^{3}, K)
and K

.

consisting of diffeomorphisms that preserve the orientations of both S^{3}

They also showed that the same is true for composite knots under a condition
Recently, Manfredi [11] gave an interesting example regarding theregarding slopes
uniqueness.

In order to state the main result, we describe free periods in another way.

knot K has period p.

Suppose a

Then we obtain a knot K'=K/\mathbb{Z}_{p} in the lens space L(p, q)=S^{3}/\mathbb{Z}_{p}for some integer q coprime to p.

Conversely, if K is a preimage of a knot K' under the102$\pi$:S^{3}\rightarrow L(p, q)

covering map

realizes a free period of p

.

,

then

Therefore,(a generator of) the deck transformation groupfocus on a knot in a lens space, especially, on

wethe fundamental group of its complement.

(Theorem 2.6). Let K' be a knot in L(p, q) with the connected preimage:=$\pi$^{-1}(K') Then the image of $\pi$_{*}:$\pi$_{1}(S^{3}\backslash K)-\rangle$\pi$_{1}(L(p, q)\backslash K') coincides with
In particular, the knot group $\pi$_{1}(S^{3}\backslash K) is a \mathrm{C}^{p} ‐group (see Defini‐\mathrm{C}^{p}($\pi$_{1}(L(p, q)\backslash K
tion 2.1).
Theorem 1
K.

As a corollary of this result, the facts mentioned above is deduced.

Corollary 1 (Corollary 3.5). A knot K in S^{3} with Out (G(K))=1 cannot be representedas

the preimage of any knot in any lens space.

Corollary 2 (Corollary 3.7). Let m, n, p\in \mathbb{Z}_{\geq 2} with \mathrm{g}\mathrm{c}\mathrm{d}(m, n)=1.

There exists

aninteger q and a knot K' in L(p, q) such that $\pi$^{-1}(K') is ambient isotopic to the torus knot

T_{m,n} or its mirror image if and only if \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1.
2

Definitions and main theorem

Definition 2.1. For a group G and p\in \mathbb{Z}_{\geq 1} , let \mathrm{C}^{p}(G) denote the subgroup of G generated

by the set \{g^{p}|g\in G\}\cup\{[g, h]|g, h\in G\} where [g, h] :=ghg^{-1}h^{-1} A group G is called.

,a\mathrm{C}^{p} ‐group if there exists a group G' such that G\cong \mathrm{C}^{p}(G').

Remark 2.2. The subgroup \mathrm{C}^{p}(G) coincides with the kernel of the composite map G\rightarrow

G_{\mathrm{a}\mathrm{b} \rightar ow G_{\mathrm{a}\mathrm{b} /pG_{\mathrm{a}\mathrm{b} .

\mathrm{C}^{2}(G) is denoted by G^{2} in [17] and by S(G) in [8]. For a prime p, \mathrm{C}^{p}(G)[16] and with the first termof the derived p‐series in [5].
Remark 2.3.

coincides with the first term of the p ‐lower central series in

Let $\pi$: $\Sigma$\rightarrow$\Sigma$' be a p‐fold cyclic covering, where $\Sigma$ is an integral homology 3‐sphere,

and K' be a knot in $\Sigma$' with the connected preimage K :=$\pi$^{-1}(K').

Remark 2.4. K is connected if and only if [K'] generates H_{1}($\Sigma$')\cong \mathbb{Z}_{p}.

The last iso‐morphism is confirmed by using the five‐term exact sequence for the short exact sequence

1\rightarrow$\pi$_{1}( $\Sigma$)\rightarrow$\pi$_{1}($\Sigma$')\rightarrow \mathbb{Z}_{p}\rightarrow 1.
Lemma 2.5. For $\Sigma$' and K'

asabove,

H_{*}($\Sigma$'\backsla hK')\cong\left\{ begin{ar y}{l \mathb {Z}&if*=0,1\ 0&otherwise \end{ar y}\right.

class represented by a meridian of K' corresponds to \pm p\in \mathbb{Z}.

The homology

103

Theorem 2.6. The image of $\pi$_{*}:$\pi$_{1}( $\Sigma$\backslash K)\mapsto$\pi$_{1}($\Sigma$'\backslash K') coincides with \mathrm{C}^{p}($\pi$_{1}($\Sigma$'\backslash K

Proof. Set G:=$\pi$_{1}( $\Sigma$\backslash K) and G' :=$\pi$_{1}($\Sigma$'\backslash K

The covering map $\pi$ induces the exact

sequence

1\rightar ow G\rightar ow^{*}G' $\pi$\rightar ow \mathbb{Z}_{p} $\psi$\rightar ow 1.
Here, $\psi$ factors through G^{\prime \mathrm{a}\mathrm{b} /pG^{\prime \mathrm{a}\mathrm{b} \cong \mathb {Z}_{p} (Lemma 2.5), and thus {\rm Im}$\pi$_{*}=\mathrm{K}\mathrm{e}\mathrm{r} $\psi$=\mathrm{C}^{p}(G')(Remark 2.2).
3\squareCorollaries
We start this section with a remark in group theory.

Remark 3.1. For a normal subgroup H of a group G\mathrm{A}\mathrm{u}\mathrm{t}(H) \mathrm{A}\mathrm{d}_{g}\mapsto \mathrm{A}\mathrm{d}_{g}|_{H},

defined if H is,is induced by definition.

the restriction map Inn (G)\rightarrow,

Furthermore, \mathrm{A}\mathrm{u}\mathrm{t}(G)\rightar ow \mathrm{A}\mathrm{u}\mathrm{t}(H) is

characteristic, that is, f(H)=H for all f\in \mathrm{A}\mathrm{u}\mathrm{t}(G).

However, therestriction map Inn (G)\rightar ow \mathrm{I}\mathrm{n}\mathrm{n}(H) is not induced in general.
The following lemma is a refinement of the well‐known factgroup H.

[13, 13.5.8] for a complete(A group G is said to be complete if the center Z(G) and the outer automorphismgroup Out (G)aretrivial.)Lemma 3.2. Let G, H be groups such that H\triangleleft G and \mathrm{A}\mathrm{d}_{g}|_{H}\in \mathrm{I}\mathrm{n}\mathrm{n}(H) for any g\in G.
Then the sequence of groups

1\rightarrow Z(H)\rightarrow $\phi$ H\times C_{G}(H)\rightarrow G $\psi$\rightarrow 1
is exact, where C_{G}(H) is the centralizer of H in G , and $\phi$(h) :=(h, h^{-1}) ,Proof. We only confirm the surjectivity of $\psi$h\in H.

.

Let

g\in G.

$\psi$(h, g) :=hg.

Then \mathrm{A}\mathrm{d}_{g}|_{H}=\mathrm{A}\mathrm{d}_{h} for someFor any h'\in H , we have[h^{-1}g, h']=h^{-1}gh'g^{-1}hh^{\prime-1}=\mathrm{A}\mathrm{d}_{h^{-1}}(\mathrm{A}\mathrm{d}_{g}(h'))h^{\prime-1}=1.
Hence, h^{-1}g\in C_{G}(H) and $\psi$(h, h^{-1}g)=g.

\squareThe next lemma is a generalization of [8, Theorem 1].

Lemma 3.3. Let G, H be as in Lemma 3.2 and suppose \mathrm{C}^{p}(G)=H,

Z(H)=1.

Then\mathrm{C}^{p}(H)=H.
Proof. By Lemma 3.2, $\psi$:H\times C_{G}(H)\rightarrow G and its restriction$\psi$|:\mathrm{C}^{p}(H)\times \mathrm{C}^{p}(K)\rightarrow \mathrm{C}^{p}(G)=H

,

(1)

104

isomorphisms, where K :=C_{G}(H) Since $\psi$(\mathrm{C}^{p}(H)\times\{1\})=\mathrm{C}^{p}(H)\mathrm{C}^{p}(K)\cong
H/\mathrm{C}^{p}(H) and thus Z(\mathrm{C}^{p}(K))=\mathrm{C}^{p}(K) On the other hand, taking the center of (1),
we have
Z(\mathrm{C}^{p}(H))\times Z(\mathrm{C}^{p}(K))\cong Z(H)=1 and Z(\mathrm{C}^{p}(K))=1 Hence, we conclude\square\mathrm{C}^{p}(K)=1 and thus $\psi$| is the identity map.
are

.

,

,

we have.

.

,

The quotient G/\mathrm{C}^{p}(G) plays a key role in our argument. The next lemma follows from

Remark 2.2 and the homomorphism theorem for the abelianization G\rightar ow G_{\mathrm{a}\mathrm{b} .
Lemma 3.4. For a group G whose abelianization is isomorphic to \mathb {Z} , the quotient group

G/\mathrm{C}^{p}(G) is isomorphic to \mathbb{Z}_{p}.
Corollary 3.5 ([6, Theorem 3.2], [3]). A knot K in S^{3}
represented as the preimage of any knot in any lens space.

with Out (G(K))=1 is not

Proof. Since Out (G(T_{m,n}))\cong \mathbb{Z}_{2} ([15]), K is not a torus knot, and thus Z(G(K))=1([3]). Hence, G(K) is complete.
Assume that there exists a knot K' in L(p, q) whose preimage is isotopic to K
we

have

G(K)=\mathrm{C}^{p}($\pi$_{1}(L(p, q)\backslash K

byLemme 3.3, we conclude \mathrm{C}^{p}G(K)=G(K).

Theorem 2.6.

Since

G(K)

.

Then

complete, byisHowever, this contradicts Lemma 3.4.

\squareIn order to prove Corollary 3.7, we quote the next lemma without proof.

Lemma 3.6 (see [12, Lemma 3.5]). Let m, n,p\in \mathbb{Z}_{\geq 1}.

If there exists a group G satisfying

and |G_{\mathrm{a}\mathrm{b}}|=mnp, then \mathrm{g}\mathrm{c}\mathrm{d}(mn, p)=1

\mathrm{C}^{p}(G)\cong \mathbb{Z}_{m}*\mathbb{Z}_{n)}G/\mathrm{C}^{p}(G)\cong \mathbb{Z}_{p}H_{*}(G) is isomorphic to H_{*}(\mathbb{Z}_{m}*\mathbb{Z}_{n}*\mathbb{Z}_{p}Corollary 3.7 ([7, Theorem 3.1]). Let m, n, p\in \mathbb{Z}_{\geq 2} with \mathrm{g}\mathrm{c}\mathrm{d}(m, n)=1.

(Moreover,

There exist an.

integer q and a knot K' in L(p, q) such that $\pi$^{-1}(K') is isotopic to the torus knot T_{m,n} or
its mirror image if and only if \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1.

Proof. If \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1

,

then

a

construction of a desired knot K'

wasgiven in [7,Theorem 3.1].

Suppose there exists K' as in the statement. We set $\pi$ í :=$\pi$_{1}(L(p, q)\backslash K')map.

The covering$\pi$:S^{3}\backslash T_{m,n}\rightarrow L(p, q)\backslash K' induces the exact sequence

1\rightarrow G(T_{m,n})=\langle a, b|a^{m}=b^{n}\}\rightarrow^{*}$\pi$_{1}' $\pi$\rightarrow \mathbb{Z}_{p}\rightarrow 1.
Since the center

Z(G(T_{m,n}))=\langle a^{m}\rangle=\mathbb{Z} is characteristic in G(T_{m,n}),thesubgroup

N:=$\pi$_{*}(\{a^{m}\rangle) of $\pi$ í is normal. We deduce the exact sequence

1\rightarrow\langle a, b|a^{m}=1=b^{n}\}\rightarrow$\pi$_{1}'$\pi$_{*}/N\rightarrow \mathbb{Z}_{p}\rightarrow 1
from the third isomorphism theorem.

By Theorem 2.6, the group G : = $\pi$ í/N satisfies\mathrm{C}^{p}(G)=\mathrm{C}^{p}($\pi$_{1}')/N=G(T_{m,n})/\langle a^{m}\}=\mathbb{Z}_{m}*\mathbb{Z}_{n}105

and G/\mathrm{C}^{p}(G)\cong \mathbb{Z}_{p}.

Hence, by Lemma 3.6, it suffices to prove |G_{\mathrm{a}\mathrm{b}}|=mnp.

The five‐term exact sequence forl \rightarrow N \rightarrow $\pi$ í \rightarrow G \rightarrow l(2)

is as follows:

0\rightarrow H_{2}(G)\rightarrow \mathbb{Z}_{G}\rightarrow \mathbb{Z}\rightarrow H_{1}(G)\rightarrow 0.
An observation on a meridian of K' proves H_{1}(G)=\mathbb{Z}_{mnp}(see [12, Corollary 3.4] fordetails).

\squareRemark 3.8. The above Hartleys result
torus links.

(Corollary 3.7) was extended by Chbili [4] toIn fact, the torus link T_{m,n} has free period p if and only if there exists an

integer q such that \mathrm{g}\mathrm{c}\mathrm{d}(p, q)=1 and p| m—nq.

Note that\mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1 implies

that the existence of such a q , however, the converse is not true without the assumption\mathrm{g}\mathrm{c}\mathrm{d}(m, n)=1.
4

Symmetric groups and braid groups
In this section,

wesuppose n\geq 3 and p\geq 2 forsimplicity. The next lemma follows

from Lemma 3.3 and the fact that \mathfrak{S}_{n} is complete for n\neq 2 , 6. Note that the case n=6

requires an additional argument (see [12, Appendix A.2]).
Lemma 4.1. The nth symmetric group \mathfrak{S}_{n} is a \mathrm{C}^{p} ‐group if and only if p is odd.
Even if a group G isa\mathrm{C}^{p} ‐group,

G/H is not necessarily a \mathrm{C}^{p} ‐group. However, thefollowing lemma assert that G/H is a \mathrm{C}^{p} ‐group for a characteristic subgroup H.
Lemma 4.2

(see [17, Theorem 1

Let G be a \mathrm{C}^{p} ‐group and f:G\rightarrow H be a surjective

homomorphism whose kernel is a characteristic subgroup of G. Then H is also a \mathrm{C}^{p} ‐group,

Proof. Suppose \mathrm{C}^{p}(G')=G.

Then we have\mathrm{C}^{p}(G'/\mathrm{K}\mathrm{e}\mathrm{r}f)=\mathrm{C}^{p}(G')/(\mathrm{K}\mathrm{e}\mathrm{r}f\cap \mathrm{C}^{p}(G') =G/\mathrm{K}\mathrm{e}\mathrm{r}f\cong H.
Hence, H is a \mathrm{C}^{p} ‐group.

Corollary 4.3.

\squareThe nth braid group B_{n} is not a \mathrm{C}^{p} ‐group for even p.

Proof. Since the nth pure braid group P_{n} :=\mathrm{K}\mathrm{e}\mathrm{r}(B_{n}\rightar ow \mathfrak{S}_{n}) is characteristic ([1, Theorem
3by Lemma 4.2, it suffices to prove that \mathfrak{S}_{n} is not a \mathrm{C}^{p} ‐group for even p

completes the proof.

.

Lemma 4.1
\square106

Remark 4.4. One of the definition of B_{n} is the fundamental group of X_{n}/\mathfrak{S}_{n} , where X_{n}is the configuration space \{(z_{1}, \ldots, z_{n})\in \mathbb{C}^{n}|z_{i}\neq z_{j}(i\neq j)\} of distinct n points in \mathbb{C},and \mathfrak{S}_{n} acts on X_{n} by permutation of coordinates.

Therefore, B_{n} is a \mathrm{C}^{p} ‐group if there

exists a topological space Y admitting a p ‐fold cyclic covering X_{n}/\mathfrak{S}_{n}\rightarrow Y and satisfying

H_{1}(X)/pH_{1}(X)=\mathbb{Z}_{p}.
Acknowledgments
The author would like to express his gratitude to Takuya Sakasai for his various sugges‐
tions.

Also, he would like to thank Makoto Sakuma for useful discussions in this topic.

Finally, this work was supported by the Program for Leading Graduate Schools, MEXT,
Japan and JSPS KAKENHI Grant Number 16\mathrm{J}07859.
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Graduate School of Mathematical Sciences
The University of Tokyo

Tokyo 153‐8914
JAPAN\mathrm{E}‐mail address:

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