# Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.1.**

- (1)
- ${G}_{1}\left(\gamma \right)={G}_{2}\left(\gamma \right)=\gamma $ for at least one component γ of Γ,
- (2)
- either Γ is non-splittable, or ${G}_{1}$ and ${G}_{2}$ are cyclic groups acting on Γ freely, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on $N\left(\Gamma \right)$.

## 2. Non-splittable Case

- (1)
- ${M}_{i}$ is an I-bundle over a compact surface and ${F}_{i}$ is the $\partial I$-subbundle,
- (2)
- ${M}_{i}$ admits a Seifert fibration in which ${F}_{i}$ is fibered,
- (3)
- $\mathrm{int}{M}_{i}$ admits a complete hyperbolic structure of finite volume, and
- (4)
- the double of $({M}_{i},{F}_{i}-\mathrm{int}{\Phi}_{i})$ along a non-empty compact submanifold ${\Phi}_{i}$ of ${F}_{i}$ is of type (3).

**Lemma**

**2.1.**

- (1)
- ${G}_{1}\left(N\left(T\right)\right)={G}_{2}\left(N\left(T\right)\right)=N\left(T\right)$,
- (2)
- ${G}_{1}$ and ${G}_{2}$ do not interchange the components of $\partial N\left(T\right)$, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on $\partial N\left(T\right)$.

**Lemma**

**2.2.**

- (1)
- ${G}_{1}\left(M\right)={G}_{2}\left(M\right)=M$ and ${G}_{1}\left(F\right)={G}_{2}\left(F\right)=F$,
- (2)
- ${G}_{1}\left(T\right)={G}_{2}\left(T\right)=T$ for at least one component T of F,
- (3)
- ${G}_{1}$ and ${G}_{2}$ induce the same permutation on the set of the components of $\partial M$, and
- (4)
- ${G}_{1}$ and ${G}_{2}$ agree on F.

- (1)
- ${\overline{G}}_{1}$ and ${\overline{G}}_{2}$ interchange ${\ell}_{2i-1}$ and ${\ell}_{2i}$ for $1\le i\le k$, and
- (2)
- ${\overline{G}}_{1}$ and ${\overline{G}}_{2}$ setwise preserve ${\ell}_{i}^{\prime}$ for $1\le i\le n$.

**Lemma**

**2.3.**

- (1)
- ${G}_{1}\left(M\right)={G}_{2}\left(M\right)=M$ and ${G}_{1}\left(F\right)={G}_{2}\left(F\right)=F$,
- (2)
- ${G}_{1}$ and ${G}_{2}$ induce the same permutation on the set of the components of $\partial M$, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on F.

**Lemma**

**2.4.**

- (1)
- ${G}_{1}\left(M\right)={G}_{2}\left(M\right)=M$ and ${G}_{1}\left(F\right)={G}_{2}\left(F\right)=F$,
- (2)
- ${G}_{1}$ and ${G}_{2}$ induce the same permutation on the set of the components of $\partial M$, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on F.

**Proposition**

**2.5.**

## 3. Possibly Splittable Case

**Lemma**

**3.1.**

- (1)
- ${G}_{1}\left(\gamma \right)={G}_{2}\left(\gamma \right)=\gamma $ for at least one component γ of Γ,
- (2)
- ${G}_{1}$ and ${G}_{2}$ are cyclic groups acting on Γ freely, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on $N\left(\Gamma \right)$.

**Lemma**

**3.2.**

- (1)
- ${G}_{1}$ and ${G}_{2}$ do not interchange the components of ${S}^{2}\times \partial I$, and
- (2)
- ${G}_{1}$ and ${G}_{2}$ agree on ${S}^{2}\times \partial I$.

**Remark**

**3.3.**

- (1)
- Suppose that $\Gamma $ is a granny knot. Then $\Gamma $ has two companion knots ${K}_{1}$ and ${K}_{2}$, both of which are trefoil knots. We obtain $E\left({K}_{1}\right)$, $E\left({K}_{2}\right)$, and a 2-fold composing space by the JSJ decomposition of $E\left(\Gamma \right)$. Figure 6 illustrates ${\mathbb{Z}}_{2}$-symmetries ${G}_{1}$ and ${G}_{2}$ of $\Gamma $ such that ${G}_{2}$ interchanges $E\left({K}_{1}\right)$ and $E\left({K}_{2}\right)$ but ${G}_{1}$ does not. By conjugating ${G}_{1}$ by a map in $\mathrm{Diff}\left({S}^{3}\right)$ which moves $N\left(\Gamma \right)$ in the longitudinal direction, ${G}_{1}$ and ${G}_{2}$ are not equivalent but agree on $\partial N\left(\Gamma \right)$. Moreover, any rational twists along incompressible tori in $E\left(\Gamma \right)$ cannot change the induced symmetries of $E\left({K}_{1}\right)$ and $E\left({K}_{2}\right)$, since the trefoil knot exterior is atoroidal.
- (2)
- Suppose that $\Gamma $ is a spatial graph which splits into non-splittable spatial graphs ${\gamma}_{1}$, ${\gamma}_{2}$ and ${\gamma}_{3}$, as illustrated in Figure 7, where ${\gamma}_{1}$ is a spatial $\theta $-curve. According to the choice of two edges of ${\gamma}_{1}$, we obtain a trefoil knot ${K}_{1}$, a figure-eight knot ${K}_{2}$, or their connected sum ${K}_{1}\#{K}_{2}$. Then any map in $\mathrm{Diff}({S}^{3},\Gamma )$ does not permute these edges. The ${\mathbb{Z}}_{2}$-symmetries ${G}_{1}$ and ${G}_{2}$ of $\Gamma $ illustrated in Figure 7 are not equivalent, since there is no map in $\mathrm{Diff}({S}^{3},\Gamma )$ which takes $\mathrm{Sing}\left({G}_{1}\right)$ to $\mathrm{Sing}\left({G}_{2}\right)$ and interchanges ${\gamma}_{2}$ and ${\gamma}_{3}$. Moreover, we cannot perform rational twists along incompressible spheres and tori in $E\left(\Gamma \right)$ to make ${G}_{2}$ equivalent to ${G}_{1}$, since any setwise ${G}_{2}$-invariant incompressible sphere in $E\left(\Gamma \right)$ separates ${\gamma}_{2}$ and ${\gamma}_{3}$.

## Acknowledgements

## References

- Simon, J. Topological chirality of certain molecules. Topology
**1986**, 25, 229–235. [Google Scholar] [CrossRef] - Flapan, E. Symmetries of Möbius ladders. Math. Ann.
**1989**, 283, 271–283. [Google Scholar] [CrossRef] - Flapan, E. Rigidity of graph symmetries in the 3-sphere. J. Knot Theor. Ramif.
**1995**, 4, 373–388. [Google Scholar] [CrossRef] - Flapan, E.; Naimi, R.; Pommersheim, J.; Tamvakis, H. Topological symmetry groups of graphs embedded in the 3-sphere. Comment. Math. Helv.
**2005**, 80, 317–354. [Google Scholar] [CrossRef] - Flapan, E.; Naimi, R.; Tamvakis, H. Topological symmetry groups of complete graphs in the 3-sphere. J. London Math. Soc.
**2006**, 73, 237–251. [Google Scholar] [CrossRef] - Noda, C. The topological symmetry group of a canonically embedded complete graph in S
^{3}. Tokyo J. Math.**1997**, 20, 45–50. [Google Scholar] [CrossRef] - Flapan, E. Infinitely periodic knots. Cana. J. Math.
**1985**, 37, 17–28. [Google Scholar] [CrossRef] - Flapan, E. The finiteness theorem for symmetries of knots and 3-manifolds with nontrivial characteristic decompositions. Topol. Appl.
**1986**, 24, 123–131. [Google Scholar] [CrossRef] - Boileau, M.; Flapan, E. Uniqueness of free actions on S
^{3}respecting a knot. Can. J. Math.**1987**, 39, 969–982. [Google Scholar] [CrossRef] - Sakuma, M. Uniqueness of symmetries of knots. Math. Z.
**1986**, 192, 225–242. [Google Scholar] [CrossRef] - Ikeda, T. Finite group actions on homologically peripheral 3-manifolds. Math. Proc. Cambridge Philos. Soc.
**2011**, 151, 319–337. [Google Scholar] [CrossRef] - Jaco, W. Lectures on three manifold topology; CBMS Regional Conference Series in Mathematic 43; American Mathematical Society: Providence, RI, USA, 1980. [Google Scholar]
- Jaco, W.; Shalen, P. Seifert fibered spaces in 3-manifolds; Memoirs of the American Mathematical Society 220: Providence, RI, USA, 1979. [Google Scholar]
- Johannson, K. Homotopy equivalences of 3-manifolds with boundaries; Lecture Notes in Mathmatics 761; Springer: Berlin, Germany, 1979. [Google Scholar]
- Morgan, J.W.; Bass, H. (Eds.) The Smith conjecture; Pure and Applied Mathematics 112; Academic Press Inc.: Orlando, FL, USA, 1984. [Google Scholar]
- Meeks, W.H.; Scott, P. Finite group actions on 3-manifolds. Invent. Math.
**1986**, 86, 287–346. [Google Scholar] [CrossRef] - Scott, P. The geometries of 3-manifolds. Bull. Lond. Math. Soc.
**1984**, 15, 401–487. [Google Scholar] [CrossRef] - Dinkelbach, J.; Leeb, B. Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom. Topol.
**2009**, 13, 1129–1173. [Google Scholar] [CrossRef] - Benedetti, R.; Petronio, C. Lectures on hyperbolic geometry; Universitext; Springer-Verlag: Berlin, Germany, 1992. [Google Scholar]
- Newman, W.H.A. A theorem on periodic transformations of spaces. Quart. J. Math.
**1931**, 2, 1–8. [Google Scholar] [CrossRef] - Meeks, W.H.; Yau, S.T. The equivariant Dehn’s lemma and loop theorem. Comment. Math. Helvetici
**1981**, 56, 225–239. [Google Scholar] [CrossRef] - Edmonds, A.L. A topological proof of the equivariant Dehn lemma. Trans. Am. Math. Soc.
**1986**, 297, 605–615. [Google Scholar] [CrossRef] - Plotnick, S.P. Finite group actions and nonseparating 2-spheres. Proc. Am. Math. Soc.
**1984**, 90, 430–432. [Google Scholar]

**Figure 4.**Modification of ${p}_{i}\left({B}_{i,k}\right)$ which makes ${p}_{i}\left({B}_{i,k}\right)$ disjoint from ${p}_{i}\left({B}_{i,j}\right)$.

© 2012 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

Ikeda, T.
Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori. *Symmetry* **2012**, *4*, 26-38.
https://doi.org/10.3390/sym4010026

**AMA Style**

Ikeda T.
Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori. *Symmetry*. 2012; 4(1):26-38.
https://doi.org/10.3390/sym4010026

**Chicago/Turabian Style**

Ikeda, Toru.
2012. "Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori" *Symmetry* 4, no. 1: 26-38.
https://doi.org/10.3390/sym4010026