An explicit relation between knot groups in lens spaces and those in $S^{3}$ (Intelligence of Low-dimensional Topology)

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(1)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 and those in S^{3}. Yuta Nozaki. Graduate School of Mathematical. Sciences,. the. University. of. Tokyo. Introduction. 1. We consider the symmetry of. article. if there exists. p\in \mathbb{Z}_{\geq 1}. knots, precisely, free periods. The details of proofs preprint [12].. be found in authors. can. f\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^{3}, K). A knot K in S^{3} is said to have. such that. f^{i}. has. ,. .. previous researches. that the torus knot. proved. example,. the trefoil. torus knot. was. Let K be. $\pi$_{1}(S^{3}\backslash K). a. T_{m,n}. on. has free. used in his. a. knot K has. periods. In [7], Hartley only if \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1 For. the existence of free. period. p if and. .. does not admit free involution. The Alexander. T_{3,2}. free period. fixed point for 0<i<p and. no. f^{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 free period p or not is interesting problem and studied by many people. We first review. in this. polynomial of. a. proof.. automorphism group Out (G(K)) of G(K)= instance, 9_{32}, 9_{33} and 24 more prime knots with 10 crossings (and. knot such that the outer. is trivial. For. images) satisfy this condition (see Kawauchi [9, Appendix F.2] or Kodama‐ [10, Table 3.1]). Then it follows from Conner‐Raymond [6, Theorem 3.2] and Burde‐Zieschang [3] that K has no free period.. their mirror. Sakuma. The purpose of this article is to deduce the above facts from. stating. our. Sakuma. [14],. knot K , if of. results,. we. .. the. on. single. result.. Before. uniqueness of free periods.. Flapan [2] independently proved that for an oriented prime 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 Boileau and. ,. \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^{3}, K) consisting. and K. previous researches. review. a. They. of. diffeomorphisms. also showed that the. regarding slopes. Recently,. same. Manfredi. that preserve the orientations of both S^{3}. is true for. [11]. gave. an. composite knots under. a. condition. interesting example regarding the. uniqueness. In order to state the main. knot K has. for. some. period p Then. integer. .. q. coprime. result,. we. we. obtain. to p. .. a. describe free knot. periods. K'=K/\mathbb{Z}_{p}. Conversely,. if K is. a. in another way.. in the lens space. preimage. of. a. Suppose. a. L(p, q)=S^{3}/\mathbb{Z}_{p}. knot K' under the.

(2) 102. covering realizes. $\pi$:S^{3}\rightarrow L(p, q). map a. free. period. of p. ,. then. Therefore,. .. the fundamental group of its. (a generator of) focus. we. on a. the deck transformation group. knot in. a. lens space,. especially,. on. 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. Corollary as. the. this. result, the facts mentioned above. (Corollary 3.5).. 1. preimage of. is deduced.. A knot K in S^{3} with Out (G(K))=1 cannot be. (Corollary 3.7). Let m, n, p\in \mathbb{Z}_{\geq 2} with \mathrm{g}\mathrm{c}\mathrm{d}(m, n)=1 integer q and a knot K' in L(p, q) such that $\pi$^{-1}(K') is ambient isotopic T_{m,n} or its mirror image if and only if \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1. Corollary. 2. There exists. .. an. to the torus knot. Definitions and main theorem. 2. Definition 2.1. For. by a. represented. any knot in any lens space.. the set. group G and p\in \mathbb{Z}_{\geq 1} , let. a. \{g^{p}|g\in G\}\cup\{[g, h]|g, h\in G\}. \mathrm{C}^{p} ‐group if there exists. Remark 2.2. The. a. where. ,. group G' such that. subgroup \mathrm{C}^{p}(G). \mathrm{C}^{p}(G). denote the. [g, h] :=ghg^{-1}h^{-1}. G\cong \mathrm{C}^{p}(G'). of G. subgroup .. generated. A group G is called. .. 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} . Remark 2.3.. \mathrm{C}^{2}(G). by G^{2}. is denoted. in. [17]. and. by S(G). coincides with the first term of the p ‐lower central series in. of the derived p‐series in Let $\pi$: $\Sigma$\rightarrow$\Sigma$' be. and K' be. a. [5].. p‐fold. a. cyclic covering,. knot in $\Sigma$' with the connected. Remark 2.4. K is connected if and. morphism. is confirmed. [8]. For a prime p, \mathrm{C}^{p}(G) [16] and with the first term. in. by using. only. where $\Sigma$ is. preimage if. K. an. integral homology 3‐sphere,. :=$\pi$^{-1}(K'). .. [K'] generates H_{1}($\Sigma$')\cong \mathbb{Z}_{p}. .. The last iso‐. 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' class. represented by. a. as. meridian. above,. H_{*}($\Sigma$'\backsla hK')\cong\left\{ begin{ar y}{l \mathb {Z}&if*=0,1\ 0&otherwise \end{ar y}\right.. of K' corresponds. to. \pm p\in \mathbb{Z}.. The. homology.

(3) 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. G^{\prime \mathrm{a}\mathrm{b} /pG^{\prime \mathrm{a}\mathrm{b} \cong \mathb {Z}_{p} (Lemma 2.5),. Here, $\psi$ factors through. and thus. {\rm Im}$\pi$_{*}=\mathrm{K}\mathrm{e}\mathrm{r} $\psi$=\mathrm{C}^{p}(G'). (Remark 2.2). 3. \square. Corollaries We start this section with. Remark 3.1. For. a. normal. \mathrm{A}\mathrm{u}\mathrm{t}(H) \mathrm{A}\mathrm{d}_{g}\mapsto \mathrm{A}\mathrm{d}_{g}|_{H} ,. defined if H is. a. remark in group. subgroup. is induced. ,. by. H of. a. theory. group G , the restriction map Inn (G)\rightarrow. definition.. characteristic, that is, f(H)=H for all f\in \mathrm{A}\mathrm{u}\mathrm{t}(G). restriction map Inn (G)\rightar ow \mathrm{I}\mathrm{n}\mathrm{n}(H) is not induced in The. following. group H.. (A. lemma is. are. Lemma 3.2. Let Then the sequence. complete. .. is. However, the. general.. refinement of the well‐known fact. a. group G is said to be. group Out (G). Furthermore, \mathrm{A}\mathrm{u}\mathrm{t}(G)\rightar ow \mathrm{A}\mathrm{u}\mathrm{t}(H). if the center. Z(G). [13, 13.5.8]. and the outer. for. a. complete. automorphism. trivial.) 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.. of groups. 1\rightarrow Z(H)\rightarrow $\phi$ H\times C_{G}(H)\rightarrow G $\psi$\rightarrow 1 is. exact, where. Proof. h\in H. We .. C_{G}(H). is the centralizer. only confirm. For any h'\in H ,. the we. of H. surjectivity of $\psi$. .. in G , and. Let. $\phi$(h) :=(h, h^{-1}) $\psi$(h, g) :=hg.. g\in G. ,. .. Then. \mathrm{A}\mathrm{d}_{g}|_{H}=\mathrm{A}\mathrm{d}_{h}. for. some. 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 The next lemma is. Lemma 3.3. Let. G,. a. $\psi$(h, h^{-1}g)=g.. generalization. H be. as. of. [8,. \square. Theorem. 1].. 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).

(4) 104. :=C_{G}(H) Since $\psi$(\mathrm{C}^{p}(H)\times\{1\})=\mathrm{C}^{p}(H) we have \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. isomorphisms,. where K. .. ,. ,. .. .. ,. The. quotient G/\mathrm{C}^{p}(G) plays. key. a. role in. argument. The. our. Remark 2.2 and the. homomorphism. Lemma 3.4. For. group G whose abelianization is. G/\mathrm{C}^{p}(G). a. isomorphic. is. Corollary. 3.5. represented. as. ([6,. the. to. Theorem. preimage of. Assume that there exists have. Lemme. isomorphic. 3.2], [3]).. a. knot K' in. K is not. L(p, q). torus. a. whose. G(K)=\mathrm{C}^{p}($\pi$_{1}(L(p, q)\backslash K by we conclude \mathrm{C}^{p}G(K)=G(K) However,. group. knot, and thus Z(G(K))=1. preimage. Corollary 3.7, Lemma. we. 3.5]).. quote the. Let m,. Since. is. to K. isotopic. G(K). .. Then. complete, by. is. this contradicts Lemma 3.4.. .. (see [12,. quotient. any knot in any lens space.. 3.3,. Lemma 3.6. to \mathb {Z} , the. A knot K in S^{3} with Out (G(K))=1 is not. Theorem 2.6.. In order to prove. G\rightar ow G_{\mathrm{a}\mathrm{b} .. \mathbb{Z}_{p}.. Proof. Since Out (G(T_{m,n}))\cong \mathbb{Z}_{2} ([15]), ([3]). Hence, G(K) is complete. we. next lemma follows from. theorem for the abelianization. next lemma without. n,p\in \mathbb{Z}_{\geq 1} If there .. |G_{\mathrm{a}\mathrm{b}}|=mnp, \mathrm{C}^{p}(G)\cong \mathbb{Z}_{m}*\mathbb{Z}_{n)}G/\mathrm{C}^{p}(G)\cong \mathbb{Z}_{p} is isomorphic to H_{*}(\mathbb{Z}_{m}*\mathbb{Z}_{n}*\mathbb{Z}_{p} and. then. exists. \square. proof. a. group G. \mathrm{g}\mathrm{c}\mathrm{d}(mn, p)=1. satisfying. (Moreover,. .. H_{*}(G). ([7,. 3.1]). Let m, n, p\in \mathbb{Z}_{\geq 2} with \mathrm{g}\mathrm{c}\mathrm{d}(m, n)=1 There exist integer q and a knot K' in L(p, q) such that $\pi$^{-1}(K') is isotopic to the torus knot T_{m,n} its mirror image if and only if \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1.. Corollary. Proof.. If. Theorem. Theorem. \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1 3.1].. Suppose map. 3.7. there exists K'. ,. as. then. .. a. construction of. a. in the statement. We set. $\pi$:S^{3}\backslash T_{m,n}\rightarrow L(p, q)\backslash K'. desired knot K'. $\pi$. was. í :=$\pi$_{1}(L(p, q)\backslash K'). .. given The. in. an or. [7,. covering. 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. N:=$\pi$_{*}(\{a^{m}\rangle). of. Z(G(T_{m,n}))=\langle a^{m}\rangle=\mathbb{Z} is characteristic in G(T_{m,n}) í is normal. We deduce the exact sequence. ,. the. subgroup. $\pi$. 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. \mathrm{C}^{p}(G)=\mathrm{C}^{p}($\pi$_{1}')/N=G(T_{m,n})/\langle a^{m}\}=\mathbb{Z}_{m}*\mathbb{Z}_{n}. : = $\pi$. í/N. satisfies.

(5) 105. G/\mathrm{C}^{p}(G)\cong \mathbb{Z}_{p} Hence, by. and. .. Lemma. 3.6, it suffices. |G_{\mathrm{a}\mathrm{b}}|=mnp.. to prove. The five‐term exact sequence for l is. as. N. \rightarrow. \rightarrow $\pi$. í. G. \rightarrow. (2). l. \rightarrow. follows:. 0\rightarrow H_{2}(G)\rightarrow \mathbb{Z}_{G}\rightarrow \mathbb{Z}\rightarrow H_{1}(G)\rightarrow 0. An observation. on. H_{1}(G)=\mathbb{Z}_{mnp} (see [12, Corollary 3.4]. meridian of K' proves. a. details).. \square. Remark 3.8. The above torus links.. In. fact,. q such that. integer. for. Hartleys. the torus link. \mathrm{g}\mathrm{c}\mathrm{d}(p, q)=1. that the existence of such. a. result. T_{m,n}. (Corollary 3.7). has free. p|. and. period. m—nq.. q , however, the. converse. by Chbili [4]. to. if there exists. an. extended. was. p if and. only. \mathrm{g}\mathrm{c}\mathrm{d}(mn,p)=1 implies. Note that. is not true without the. assumption. \mathrm{g}\mathrm{c}\mathrm{d}(m, n)=1. 4. Symmetric In this. section,. we. groups and braid groups suppose. requires. additional argument. an. Lemma 4.1. The nth Even if. following. a. (see [17,. G/H. homomorphism. group. Proof. Suppose \mathrm{C}^{p}(G')=G. .. a. complete. is. \mathfrak{S}_{n}. G/H a. Theorem 1. whose kernel is. is. for. n\neq 2. ,. 6. Note that the. case. n=6. (see [12, Appendix A.2]).. \mathrm{C}^{p} ‐group,. lemma assert that. Lemma 4.2. \mathfrak{S}_{n}. symmetric. group G is. a. next lemma follows. n\geq 3 and p\geq 2 for simplicity. The. from Lemma 3.3 and the fact that. is. is not. Let G be. a. we. if. necessarily. \mathrm{C}^{p} ‐group for. characteristic. Then. \mathrm{C}^{p} ‐group. a. a. and a. only if p. However, the. \mathrm{C}^{p} ‐group.. characteristic. subgroup. H.. be. a. surjective. of G. Then H is also. a. \mathrm{C}^{p} ‐group,. \mathrm{C}^{p} ‐group and. subgroup. is odd.. f:G\rightarrow H. 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. Corollary. by. \mathrm{C}^{p} ‐group.. 4.3.. Proof. Since 3. a. The nth braid group. B_{n}. the nth pure braid group P_{n}. Lemma. completes. \square. the. 4.2, it suffices. proof.. is not. a. \mathrm{C}^{p} ‐group. :=\mathrm{K}\mathrm{e}\mathrm{r}(B_{n}\rightar ow \mathfrak{S}_{n}). to prove that. \mathfrak{S}_{n} is. not. a. for. even. p.. is characteristic. \mathrm{C}^{p} ‐group for. even. p. ([1, .. Theorem. Lemma 4.1 \square.

(6) 106. Remark 4.4. One of the definition of is the. and. configuration. \mathfrak{S}_{n}. exists. a. acts. on. is the fundamental group of. \{(z_{1}, \ldots, z_{n})\in \mathbb{C}^{n}|z_{i}\neq z_{j}(i\neq j)\}. space. X_{n} by permutation. topological. B_{n}. space Y. of coordinates.. admitting. a. p ‐fold. X_{n}/\mathfrak{S}_{n}. of distinct. Therefore, B_{n}. is. a. n. ,. where. points. in. X_{n}. \mathbb{C},. \mathrm{C}^{p} ‐group if there. 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 tions.. Also,. Finally, Japan. gratitude. to. Takuya Sakasai for. his various sugges‐. he would like to thank Makoto Sakuma for useful discussions in this. this work. supported by. was. the. Program. for. Leading. Graduate. topic.. Schools, MEXT,. and JSPS KAKENHI Grant Number 16\mathrm{J}07859.. References. [1]. E. Artin. Braids and. [2]. M. Boileau and E.. J.. [3]. G. Burde and H.. N. Chbili. A. on. S^{3} respecting. a. knot. Canad.. Eine. Zieschang.. Kennzeichnung. der Torusknoten.. Math.. Ann.,. 1966. new. criterion for knots with free. T. Cochran and S.. Soc.. [6]. of free actions. ,. (6), 12(4):465-477. [5]. Flapan. Uniqueness. 1947.. Math., 39(4):969-982 1987.. 167:169−176,. [4]. permutations. Ann. of Math. (2), 48:643−649,. ,. periods. Ann.. Fac. Sci. Toulouse Math.. 2003.. Harvey. Homology. (2), 78(3):677-692. P. E. Conner and F.. ,. and derived p ‐series of groups. J. Lond. Math.. 2008.. Raymond. Manifolds. with few. periodic homeomorphisms.. In. Compact Transformation Groups (Univ. Conference Massachusetts, Amherst, Mass., 1971), Part II, pages 1‐75. Lecture Notes in Math., Proceedings of Vol. 299.. Springer, Berlin,. [7]. R.. Hartley.. [8]. M.. Haugh. A.. on. 1972.. Knots with free. period. Canad.. and D. MacHale. The. Acad. Sect.. [9]. the Second. A, 97(2):123-129. Kawauchi, (ed.). A. ,. survey. J.. Math., 33(1):91-102. subgroup generated by. ,. 1981.. the squares. Proc.. Roy. Irish. 1997.. of knot theory.. lated and revised from the 1990. Birkhäuser. Japanese original by. Verlag, Basel,. the author.. 1996. Trans‐.

(7) 107. [10]. K. Kodama and M. Sakuma.. Knots 90. [11]. (Osaka, 1990),. E. Manfredi. Lift in the. Symmetry. 3‐sphere. Ramifications, 23(5):1450022 21, ,. [12]. Y. Nozaki. An. 2016,. [13]. groups of. pages 323‐340. de. prime knots. Gruyter, Berlin,. up to 10. crossings.. In. 1992.. of knots and links in lens spaces. J. Knot. Theory. 2014.. explicit relation between. knot groups in lens spaces and those in. S^{3},. arXiv: 1602.05884.. D. J. S. Robinson. A. Mathematics.. [14]. M. Sakuma.. [15]. O. Schreier.. course. in the. Springer‐Verlag, Uniqueness. Über. of. theory of groups,. New. volume 80 of Graduate Texts in. York, second edition,. 1996.. symmetries of knots. Math. Z., 192(2):225-242 1986. ,. die gruppen A^{a}B^{b}=1. .. Abh. Math. Sem. Univ.. Hamburg, 3(1):167-. 169, 1924.. [16]. J.. [17]. H. S. Sun.. Stallings. Homology On groups. and central series of groups. J.. generated by the. squares.. Algebra, 2:170−181,. Fibonacci. 1965.. Quart., 17(3):241-246,. 1979.. Graduate School of Mathematical Sciences The. University of Tokyo. Tokyo. 153‐8914. JAPAN \mathrm{E}‐mail address:. [email protected]‐tokyo.ac.jp \displayst le\ovalbox{\t smal REJ CT}\overline{\nearow5_{\backsla h\star\neq\star$\beta$_{$\pi$\mathscr{X}$\iota$\ovalbox{\t smal REJ CT}\ovalbox{\t smal REJ CT}_{\backsla h}\mathrm{J}^{\backsla h}\mp\mathrm{f}\mathrm{f}\mathrm{l}_J\mathrm{t}^{*\ovalbox{\t smal REJ CT}_{\backsla h}\mathrm{J}^{\backsla h} ^{B}^{\mathrm{J} \grave{r}^{\backsla h}\rightarow\frac{\backsla h\backsla h}{\neq}\grave{$\Gam a$}^{\backsla h}\rightarow. IEIfi. r_{\mathrm{A} \not\in*.

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