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Symmetry 2012, 4(1), 26-38; https://doi.org/10.3390/sym4010026

Article
Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori
Department of Mathematics, Faculty of Science, Kochi University, 2-5-1 Akebono-cho, Kochi-Shi, Kochi 780-8520, Japan
Received: 14 November 2011; in revised form: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012

Abstract

:
A symmetry group of a spatial graph Γ in S 3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S 3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ .
Keywords:
3-manifold; geometric topology; symmetry; finite group action; spatial graph; rational twist

1. Introduction

There are several approaches to the theory of graphs embedded in the 3-sphere, which are often motivated by molecular chemistry, since the chemical properties of a molecule depend on the symmetries of its molecular bond graph (see, for example, [1]). The symmetries of an abstract graph Γ are described by automorphisms. If Γ is embedded in S 3 , some of these automorphisms are induced from self-diffeomorphisms of S 3 . For example, [2,3,4,5,6] studied the extendabilities of the automorphisms of Γ , mainly in the case of Möbius ladders, complete graphs, and 3-connected graphs.
Even if the automorphisms of Γ extend to self-diffeomorphisms of S 3 , we face the problem of the uniqueness of the extensions. In this situation, it is enough to consider Γ to be a topological space, since we need to study self-diffeomorphisms of S 3 which agree on Γ . In the case of a non-torus knot in S 3 , there are only finitely many conjugacy classes of symmetries (see [7,8]). For a cyclic period or a free period of a knot in S 3 , it is shown in [9,10] that the cyclic group generated by the periodic self-diffeomorphism of S 3 defining the symmetry is unique up to conjugate in some cases. Moreover, the author [11] generalized this result to the case of links in S 3 . In this paper, we generalize these results to the case of symmetries of spatial graphs in S 3 .
Suppose that any component of Γ is a non-trivial graph with no leaf. We see Γ as a geometric simplicial complex, and denote by | Γ | the underlying topological space of Γ . A tame embedding of | Γ | into S 3 is called a spatial embedding of Γ into S 3 , or simply a spatial graph Γ in S 3 . We say that Γ is splittable if there exists a sphere in S 3 disjoint from Γ that separates the components of Γ . We say that Γ is non-splittable if it is not splittable. Suppose that an incompressible torus in S 3 Γ bounds a solid torus V in S 3 containing Γ . The core of V is called a companion knot of Γ if it is not ambient isotopic to Γ in V. If there is no companion knot of Γ , every incompressible torus in S 3 Γ separates the components of Γ .
Let M be a 3-manifold, and X a submanifold of M. Denote by N ( X ) a regular neighborhood of X, and by E ( X ) = M int N ( X ) the exterior of X. We refer to a finite subgroup G of the diffeomorphism group Diff ( M ) as a finite group action on M. Finite group actions G 1 and G 2 on M are equivalent (relative to X) if some h Diff ( M ) conjugates G 1 to G 2 (and restricts to the identity map on X). A symmetry group G of a spatial graph Γ in S 3 is a finite group action on the pair ( S 3 , Γ ) which preserves the orientation of S 3 .
Let S 2 be the unit sphere in R 3 , and S 1 the unit circle in the x y -plane in R 3 . Denote by Rot θ Diff ( R 3 ) the rotation about the z-axis through angle θ . Suppose that σ n Diff ( S 2 × I ) and τ n Diff ( S 1 × S 1 × I ) , where n R , is given by σ n ( x , t ) = ( Rot 2 π n t ( x ) , t ) and τ n ( x , y , t ) = ( Rot 2 π n t ( x ) , y , t ) . Let F be a 2-sided sphere or torus embedded in a 3-manifold M. Split M open along F into a (possibly disconnected) 3-manifold M F . Denote by F and F + the boundary components of M F originated from F. An n-twist along F is a discontinuous map on M induced from a diffeomorphism on M F F which restricts to the identity map on E ( F + ) and the map on N ( F + ) conjugate to σ n or τ n according as F is a sphere or not. We say that the n-twist is rational if n Q . Figure 1 illustrates a rotational symmetry of S 3 with a setwise invariant sphere S, and its conjugate by a 1 / 2 -twist along S.
Our main theorem is the following:
Theorem 1.1.
Let Γ be a spatial graph in S 3 with no companion knot. Suppose that G 1 and G 2 are symmetry groups of Γ such that
(1)
G 1 ( γ ) = G 2 ( γ ) = γ 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 ( Γ ) .
Then there is a finite sequence of rational twists along incompressible spheres and tori in E ( Γ ) whose composition conjugates G 2 to a symmetry group of Γ equivalent to G 1 relative to N ( Γ ) .
This paper is arranged as follows. In Section 2, we study symmetry groups of non-splittable spatial graph in terms of the equivariant JSJ decomposition of the exteriors. In Section 3, we establish a canonical version of the equivariant sphere theorem for the exteriors of spatial graphs with cyclic symmetry groups, and prove Theorem 1.1.

2. Non-splittable Case

For a non-splittable spatial graph Γ in S 3 with a non-trivial symmetry group, there is a canonical method for splitting E ( Γ ) equivariantly into geometric pieces by the loop theorem, the Dehn’s lemma, and the JSJ decomposition theorem (see [12,13,14]).
Let M be a Haken 3-manifold with incompressible boundary. The JSJ decomposition theorem and Thurston’s uniformization theorem [15] assert that there is a canonical way of splitting the pair ( M , M ) along a disjoint, non-parallel, essential annuli and tori into pieces ( M i , F i ) each of which is one of the following four types:
(1)
M i is an I-bundle over a compact surface and F i is the I -subbundle,
(2)
M i admits a Seifert fibration in which F i is fibered,
(3)
int M i admits a complete hyperbolic structure of finite volume, and
(4)
the double of ( M i , F i int Φ i ) along a non-empty compact submanifold Φ i of F i is of type (3).
For a finite group action G on M, the fixed point set Fix ( G ) of G is the set of points in M each of which has the stabilizer G. The singular set Sing ( G ) of G is the set of points in M each of which has a non-trivial stabilizer.
Lemma 2.1.
Let T be a torus embedded in S 3 . Suppose that G 1 and G 2 are orientation-preserving finite group actions on S 3 such that
(1)
G 1 ( N ( T ) ) = G 2 ( N ( T ) ) = N ( T ) ,
(2)
G 1 and G 2 do not interchange the components of N ( T ) , and
(3)
G 1 and G 2 agree on N ( T ) .
Then a rational twist along a component of N ( T ) conjugates G 2 to a finite group action G ^ 2 on S 3 such that the actions of G 1 and G ^ 2 on N ( T ) are equivalent relative to N ( T ) .
Proof. 
It is enough by Lemma 2.4 of [11] to consider the case where the actions of G 1 and G 2 on N ( T ) are not free. For each G i , Theorem 2.1 of [16] implies that N ( T ) T × I admits a G i -invariant product structure P i , in which Sing ( G i ) N ( T ) consists of I-fibers. Since each element of G i takes a meridian of T to a meridian of T, the setwise stabilizer of each I-fiber is a trivial group or a 2-fold cyclic group. Therefore, the quotient space N ( T ) / G i admits the induced I-bundle structure over a 2-orbifold B with underlying surface F and n cone points of index two. Since T is a torus, the orbifold Euler characteristic χ orb ( B ) of B is calculated as follows (see [17]):
χ orb ( B ) = χ ( F ) n / 2 = 0 .
Since n > 0 , F is a sphere and n = 4 holds.
Denote by p i : N ( T ) N ( T ) / G i the projection map onto the quotient space for each i, and by T t the T-fiber T × { t } in P 1 . Connect the four cone points on p 1 ( T 0 ) cyclically by a collection of arcs a ¯ 1 , a ¯ 2 , a ¯ 3 , and a ¯ 4 with disjoint interiors. Each a ¯ i lifts to an essential loop a i on T 0 such that a i and a j with i j are disjoint if | j i | = 2 , and otherwise a i meets a j transversally in a point. Suppose that each a i is isotopic to a loop b i on T 1 along an annulus B i saturated by I-fibers in P 1 , and to a loop c i on T 1 along an annulus C i saturated by I-fibers in P 2 . Then the endpoints of each p 2 ( c i ) is connected by p 2 ( b j ) with | i j | = 0 or 2. Since the underlying surface of p 2 ( T 1 ) is a sphere, i = 1 4 p 2 ( c i ) is isotopic to i = 1 4 p 2 ( b i ) relative to the cone points. Therefore, G 2 ( i = 1 4 C i ) is moved by an G 2 -equivariant isotopy relative to T 0 so as to agree with G 1 ( i = 1 4 B i ) on T 1 .
The I-bundle structures in P 2 and P 1 respectively induce orbifold isomorphisms φ 1 : p 2 ( T 1 ) p 2 ( T 0 ) and φ 2 : p 2 ( T 0 ) p 2 ( T 1 ) such that h ¯ = φ 2 φ 1 setwise preserves the loop i = 1 4 p 2 ( b i ) . The restriction of h ¯ on i = 1 4 p 2 ( b i ) is isotopic relative to the cone points to the identity map or an involution. Since i = 1 4 p 2 ( b i ) splits p 2 ( T 1 ) into two disks with no cone point, P 2 is deformed by a G 2 -equivariant isotopy so that afterwards h ¯ is the identity map or an involution.
Take an h ¯ -invariant S 1 -bundle structure S 1 on p 2 ( T 1 ) p 2 ( b 1 b 3 ) with respect to which p 2 ( b 2 ) and p 2 ( b 4 ) are cross sectional, and an h ¯ -invariant S 1 -bundle structure S 2 on p 2 ( T 1 ) p 2 ( b 2 b 4 ) with respect to which every fiber in S 1 splits into two cross sections. Then S 1 and S 2 induce a G 2 -invariant product structure S 1 × S 1 on T 1 . Let h : T 1 T 1 be the lift of h ¯ which takes each c i to b i . Then we have h = Rot 2 π m × Rot 2 π n for some rational numbers m and n.
Assume ( m , n ) ( 0 , 0 ) . Take a rational number γ so that γ m and γ n are coprime integers. Then α γ m + β γ n = 1 holds for some integers α and β . Let ρ : R 2 S 1 × S 1 be the covering map given by ρ ( x , y ) = ( Rot 2 π x ( 1 , 0 ) , Rot 2 π y ( 1 , 0 ) ) . Denote by φ the linear transformation on R 2 represented by α β γ m γ n . Then the map ρ φ ρ 1 Diff ( S 1 × S 1 ) conjugates h to Rot 2 π / γ × id S 1 . Thus, h extends to 1 / γ -twist τ along T 1 . Since h conjugates the action of G 2 on T 1 to itself, τ conjugates G 2 to a finite subgroup of Diff ( S 3 ) . Therefore, it is enough to consider the case ( m , n ) = ( 0 , 0 ) .
It is obvious that h = Rot 2 π k × Rot 2 π l holds for any integers k and l. By verifying that, for some choice of k and l, the above argument applied to Rot 2 π k × Rot 2 π l makes G 2 ( i = 1 4 C i ) isotopic to G 1 ( i = 1 4 B i ) relative to N ( T ) , we may assume that they agree.
By considering an isotopy of N ( T ) relative to N ( T ) which takes P 2 to P 1 on Sing ( G 1 ) N ( T ) , we may assume that G 1 and G 2 agree on Sing ( G 1 ) N ( T ) . Note that Sing ( G 1 ) N ( T ) splits G 1 ( i = 1 4 B i ) into disks, and that G 1 ( i = 1 4 B i ) splits N ( T ) into balls. Then the identity map on p 2 ( Sing ( G 2 ) N ( T ) ) extends to an orbifold isomorphism ψ : p 2 ( i = 1 4 C i ) p 1 ( i = 1 4 B i ) . Since the quotient space of any finite group action on D 3 is isomorphic to one of the orbifolds listed on page 191 of [15], ψ and the identity map on p 2 ( N ( T ) ) extend to an orbifold isomorphism p 2 ( N ( T ) ) p 1 ( N ( T ) ) . Thus, G 1 and G 2 are equivalent relative to N ( T ) . Hence, the conclusion follows. □
Lemma 2.2.
Let M be a Seifert manifold in S 3 with non-empty boundary, and F a non-empty closed submanifold of M . Suppose that G 1 and G 2 are finite group actions on S 3 such that
(1)
G 1 ( M ) = G 2 ( M ) = M and G 1 ( F ) = G 2 ( F ) = F ,
(2)
G 1 ( T ) = G 2 ( T ) = 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 M , and
(4)
G 1 and G 2 agree on F.
Then there is a finite sequence of rational twists along incompressible tori in M whose composition conjugates G 2 to a finite group action G ^ 2 on S 3 such that the actions of G 1 and G ^ 2 on M are equivalent relative to F.
Proof. 
The case M = D 2 × S 1 and F = M , the case M = S 1 × S 1 × I and F = M , and the case M = S 1 × S 1 × I and F M respectively follow from Lemma 2.1 of [11], Lemma 2.1 of this paper, and Theorem 8.1 of [16]. We therefore exclude these cases.
Denote by k ξ k the system of the exceptional fibers ξ k in M. Let N ( ξ k ) be a fibered regular neighborhood of each ξ k . It follows from Theorem 2.2 of [16] that each G i preserves some Seifert fibration S i of M. Then the uniqueness of a Seifert fibration of M (see VI.18.Theorem of [12]) implies that k N ( ξ k ) is isotopic to a setwise G i -invariant fibered regular neighborhood of the system of exceptional fibers in S i . Since Lemma 3.1 of [11] implies that the orders of the exceptional fibers are pairwise coprime, we may assume that G 1 ( N ( ξ k ) ) = G 2 ( N ( ξ k ) ) = N ( ξ k ) for each k. Therefore, it is enough by Lemma 2.1 of [11] to consider the case where M is a product S 1 -bundle.
It follows from Theorem 2.1 of [16] that M admits a G 1 -invariant product structure P 1 . If F = M , M admits a G 2 -invariant product structure P 2 which agrees with P 1 on F (see Theorem 2.3 of [16]). If F M , we see M as the quotient of the double M ¯ of M along M F by Z 2 generated by an orientation-reversing involution, and apply the same argument to the finite group action on M ¯ , which is the extension of Z 2 by G 2 . Then we obtain a G 2 -invariant product structure P 2 of M which agrees with P 1 on F.
By the uniqueness of the S 1 -bundle structure of M (see VI.18.Theorem of [12]), there is a map φ Diff ( M ) isotopic to the identity which takes the S 1 -bundle structure induced by P 1 to the S 1 -bundle structure induced by P 2 . Modify φ in P 2 by a fiber preserving isotopy in a fibered regular neighborhood of F so as to restrict to the identity map on F. By conjugating G 2 by φ , we may therefore assume that P 1 and P 2 induce the same S 1 -bundle structure of M.
Let p : M B be the projection map onto the base surface B. Each G i induces a finite group action G ¯ i on B. We consider B to be lying on S 2 . Then each G ¯ i extends to an action on S 2 . Since G ¯ 1 and G ¯ 2 agree on p ( F ) , the quotient spaces B / G ¯ 1 and B / G ¯ 2 are orbifold isomorphic to suborbifolds of the same spherical orbifold listed on page 188 of [15]. We may assume that G ¯ 1 and G ¯ 2 are not orientation-preserving, otherwise the conclusion follows from Lemma 3.2 and Remark 3.3 of [11]. Then the assumption G 1 ( T ) = G 2 ( T ) = T implies that each G ¯ i is generated by the reflection of S 2 in a loop. Since G 1 and G 2 permute the components of M similarly, B consists of loops 1 , , 2 k , 1 , , n such that
(1)
G ¯ 1 and G ¯ 2 interchange 2 i 1 and 2 i for 1 i k , and
(2)
G ¯ 1 and G ¯ 2 setwise preserve i for 1 i n .
Without loss of generality, 1 = p ( T ) . Denote by Fix ( G ¯ i ) the fixed point circle of the action of each G ¯ i on S 2 . Suppose that each Fix ( G ¯ i ) is equipped with an orientation, and splits B into two pieces B i , 1 and B i , 2 so that 1 B 1 , 1 = 1 B 2 , 1 and 1 B 1 , 2 = 1 B 2 , 2 . We may assume without loss of generality that 2 i 1 B 1 , 1 and 2 i B 1 , 2 for 1 i k , and that we meets 1 , , n in order as we go along Fix ( G ¯ 1 ) .
Suppose 2 i 1 B 2 , 2 and 2 i B 2 , 1 for some i. By taking a proper arc on B / G ¯ 2 connecting 2 i / G ¯ 2 and Fix ( G ¯ 2 ) / G ¯ 2 , we obtain a setwise G ¯ 2 -invariant arc α on B which meets Fix ( G ¯ 2 ) in a point and connects 2 i 1 and 2 i . Then Fix ( G ¯ 2 ) is modified by the half twist along the loop N ( 2 i 1 2 i α ) int B , denoted by λ , so that afterwards 2 i 1 B 2 , 1 and 2 i B 2 , 2 , as illustrated in Figure 2. The argument presented for the proof of Lemma 2.1 implies that this modification is realized by a 1 / 2 -twist along the torus p 1 ( λ ) which conjugates G 2 to a subgroup of Diff ( S 3 ) . We may therefore assume 2 i 1 B 2 , 1 and 2 i B 2 , 2 for 1 i k .
Suppose that i and j are connected by an arc α in Fix ( G ¯ 2 ) B . Then Fix ( G ¯ 2 ) is modified by the half twists along the loop λ = N ( i j α ) int B so as to meet i and j in the reverse order, as illustrated in Figure 3, which is realized by the conjugation of G 2 by a 1 / 2 -twist along the torus p 1 ( λ ) , as before. Since every permutation on the set { 2 , , n } is a product of transpositions, we may assume that Fix ( G ¯ 2 ) meets 1 , , n in order. Moreover, we can change the order in which Fix ( G ¯ 2 ) meets the two points in i Fix ( G ¯ 2 ) by the half twists along i , which is also realized by a 1 / 2 -twist along the torus p 1 ( i ) . We may therefore assume that G ¯ 2 is equivalent to G ¯ 1 relative to B .
Now we may assume G ¯ 1 = G ¯ 2 . Take a map h Diff ( M ) which restricts to the identity map on F and takes P 2 to P 1 setwise preserving every S 1 -fiber. It is easy to verify that h is extendable to a map in Diff ( S 3 ) . Hence, the conclusion follows by conjugating G 2 by h. □
Lemma 2.3.
Let M be a compact connected 3-manifold in S 3 with non-empty boundary whose interior admits a complete hyperbolic structure of finite volume, and F a non-empty closed submanifold of M . Suppose that G 1 and G 2 are finite group actions on S 3 such that
(1)
G 1 ( M ) = G 2 ( M ) = M and G 1 ( F ) = G 2 ( F ) = F ,
(2)
G 1 and G 2 induce the same permutation on the set of the components of M , and
(3)
G 1 and G 2 agree on F.
Then there is a sequence of rational twists along tori in F whose composition conjugates G 2 to a finite group action G ^ 2 such that the actions of G 1 and G ^ 2 on M is equivalent relative to F.
Proof. 
It follows from Theorem 5.5 of [18] that int M admits two complete hyperbolic structures of finite volume, one is G 1 -invariant and the other is G 2 -invariant. Mostow’s rigidity theorem [15] implies that complete hyperbolic structures of finite volume on int M are unique up to isometry representing the identity map on Out ( π 1 ( M ) ) . We may therefore assume that int M is endowed with the G 1 -invariant hyperbolic structure, and that G 2 is conjugate to an isometric action G 2 by h Diff ( M ) which is isotopic to the identity map.
Next, we are going to modify h in a regular neighborhood of F so as to restrict to the identity map on F. It follows from Propostition D.3.18 of [19] that F consists of tori. Let h t be an isotopy from h to the identity map. Denote by G ¯ 2 the finite group action on F × I whose restriction on F × { t } is induced from the finite group action on F given by the conjugate of G 2 by h t . In particular, the actions of G ¯ 2 on F × { 0 } and F × { 1 } are respectively given by G 2 and G 2 . Note that G ¯ 2 preserves the product structure F × I , and that we can embed F × I in S 3 so that G ¯ 2 extends to a finite group action on S 3 .
We consider the partition of the set of the components of F into the orbits under the permutation induced by G 2 . Suppose that the orbits are represented by T 1 , , T n . Lemma 2.1 implies that a rational twist along T i × { 1 } conjugates the setwise stabilizer of T i × I in G ¯ 2 so that the action on T i × I is equivalent relative to T i × I to the action which preserves the product structure. Suppose that the rational twists along the tori in F × { 1 } are equivariantly induced from those along T 1 × { 1 } , , T n × { 1 } . By conjugating G ¯ 2 by their composition, it is equivalent relative to F × I to the action which preserves the product structure. This implies that h is modified equivariantly so as to restrict to the identity map on F.
Suppose that g 1 G 1 and g 2 G 2 agree on F. Then g 1 g 2 1 restricts to the identity map on F. Since the isometry group of int M is finite (see [15]), Newman’s theorem [20] implies g 1 = g 2 . Hence, G 1 and G 2 agree on M. This completes the proof. □
Lemma 2.4.
Let M be a compact connected 3-manifold in S 3 with non-empty boundary such that the double M ¯ of M along a non-empty compact submanifold Φ of M admits a complete hyperbolic structure of finite volume in its interior. Let F be a closed submanifold of M containing Φ. Suppose that G 1 and G 2 are finite group actions on S 3 such that
(1)
G 1 ( M ) = G 2 ( M ) = M and G 1 ( F ) = G 2 ( F ) = F ,
(2)
G 1 and G 2 induce the same permutation on the set of the components of M , and
(3)
G 1 and G 2 agree on F.
Then there is a finite sequence of rational twists along tori in F whose composition conjugates G 2 to a finite group action G ^ 2 such that the actions of G 1 and G ^ 2 on M are equivalent relative to F.
Proof. 
We see M as the quotient of M ¯ by Z 2 generated by an orientation-reversing involution. Each G i induces a finite group action G ¯ i on M ¯ which is an extension of Z 2 by G i . As in the proof of Lemma 2.3, we consider int M ¯ endowed with a G ¯ 1 -invariant hyperbolic structure. Then some h ¯ Diff ( M ¯ ) , which is isotopic to the identity map, conjugates G ¯ 2 to an isometric action G ¯ 2 . Clearly, Φ meets int M ¯ in a totally geodesic surface, and therefore h ¯ ( Φ ) = Φ holds.
Suppose that g ¯ 1 G ¯ 1 and g ¯ 2 G ¯ 2 respectively induce g 1 G 1 and g 2 G 2 which agree on F. Then g ¯ 1 1 g ¯ 2 restricts to an isometry on each component Φ i of Φ , which is a compact surface of negative Euler characteristic (see Propostition D.3.18 of [19]). Since g ¯ 1 1 g ¯ 2 is trivial in Out ( π 1 ( Φ i ) ) , g ¯ 1 and g ¯ 2 agree on Φ i . Therefore, [20] implies g 1 = g 2 . Hence, some h Diff ( M ) , which setwise preserves Φ and is isotopic to the identity map, conjugates the action of G 1 on M to G 2 .
It follows from Proposition D.3.18 of [19] that F Φ consists of tori. As in the proof of Lemma 2.3, modify h in N ( F Φ ) by rational twists along tori in F Φ so that afterwards h restricts to the identity map on F Φ and conjugates the action of G 1 on M to G 2 . Moreover, we may assume by Lemma 2.3 of [11] that h restricts to the identity map on Φ . Since h extends to an automorphism of S 3 which is diffeomorphic outside M, the conclusion follows. □
Proposition 2.5.
Theorem 1.1 is true, if Γ is non-splittable.
Proof. 
The equivariant loop theorem (see Chapter VII of [15] and [21]) implies that there is a G 1 -invariant system D 1 of disjoint disks properly embedded in E ( Γ ) which splits E ( Γ ) into pieces with incompressible boundary. The equivariant Dehn’s lemma [21,22] implies that the boundary loops of D 1 bound a G 2 -invariant system D 2 of disjoint disks properly embedded in E ( Γ ) . Since Γ is non-splittable, E ( Γ ) is irreducible. Therefore, there is an isotopy of E ( Γ ) relative to E ( Γ ) which takes D 2 to D 1 . Since any finite group action on D 2 is orthogonal [15], we may assume that G 1 and G 2 agree on D 1 . Moreover, the induced actions on the balls obtained by splitting E ( Γ ) along D 1 are equivalent relative to the boundary (see [15]). Therefore, it is enough to consider the case where E ( Γ ) is a Haken manifold with incompressible boundary.
We may assume by the equivariant JSJ decomposition theorem (see Theorem 8.6 of [16]) and by the uniqueness of the JSJ decomposition [13,14] that there is a G 1 -invariant and G 2 -invariant system T of essential annuli and tori in E ( Γ ) realizing the canonical JSJ decomposition of the pair ( E ( Γ ) , E ( Γ ) ) .
The argument presented for the proof of Proposition 3.10 of [11] implies that some h Diff ( S 3 ) , which is isotopic to the identity map relative to N ( Γ ) , conjugates G 2 to a finite group action which agree with G 1 on the annuli in T . We may therefore assume that T contains no annuli.
The rest of the proof proceeds by induction on the number of tori in T . Take a piece M k attaching E ( Γ ) . By Lemmas 2.2, 2.3 and 2.4, it is enough to consider the case where G 2 agrees with G 1 on G 1 ( M k ) . Moreover, we may assume by Lemma 2.1 that G 1 and G 2 agree on the components of cl ( E ( Γ ) G 1 ( M k ) ) each of which is a product I-bundle over a torus. Hence, the conclusion follows by the induction hypothesis. □

3. Possibly Splittable Case

For a symmetry group G of a splittable spatial graph Γ in S 3 , there is a setwise G-invariant system S of spheres realizing the prime factorization of E ( Γ ) (see [23]). However, S is not unique in contrast to the JSJ decomposition of a Haken 3-manifold. If some component of Γ is setwise invariant and every essential sphere in E ( Γ ) has a trivial stabilizer, there is a canonical choice of S (see [11]). We first prove that this is possible also in the setting of Theorem 1.1.
Lemma 3.1.
Let Γ be a splittable spatial graph in S 3 . Suppose that G 1 and G 2 are symmetry groups of Γ such that
(1)
G 1 ( γ ) = G 2 ( γ ) = γ 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 ( Γ ) .
Then each G i admits a setwise G i -invariant system B i of disjoint balls in S 3 not containing γ such that each B i realizes the prime factorization of E ( Γ ) . Moreover, for some choice of B 1 and B 2 , there is a finite sequence of rational twists along incompressible tori in E ( Γ ) int B 2 and a map in Diff ( S 3 ) which restricts to the identity map on N ( Γ ) whose composition conjugates the action of G 2 on S 3 int B 2 to the action of G 1 on S 3 int B 1 .
Proof. 
Denote by Γ γ the non-splittable spatial subgraph of Γ containing γ which is obtained by the prime factorization of E ( Γ ) . It follows from the equivariant sphere theorem [23] that each G i admits a setwise G i -invariant system S i = S i , 1 S i , n of disjoint, non-parallel, essential spheres in E ( Γ ) realizing the prime factorization. Suppose that each S i , j bounds a ball B i , j disjoint from γ . Note that Sing ( G i ) avoids B i , j or meets B i , j in a trivial 1-string tangle (see [15]).
Suppose B i , j B i , k for some distinct j and k. Denote by p i : S 3 S 3 / G i = S 3 the projection map onto the quotient space. Take an arc α properly embedded in p i ( B i , k int B i , j ) which connects p i ( B i , k ) and p i ( B i , j ) . Suppose that α lies on p i ( Sing ( G i ) ) if Sing ( G i ) connects B i , j and B i , k . By replacing B i , j with another ball in int B i , k if necessary, α meets S i in its endpoints. By drilling into p i ( B i , k ) along α p i ( B i , j ) , B i , k is deformed to a ball disjoint from B i , j , as illustrated in Figure 4 in which the result of the deformation is presented in a cross-sectional view. By a finite repetition of this operation, we obtain a system B i = B i , 1 B i , n of disjoint balls. This proves the first half of the lemma. Without loss of generality, Γ B 2 , j = Γ B 1 , j for 1 j n .
Proposition 2.5 implies that there is a finite sequence of rational twists along incompressible tori in E ( Γ γ ) whose composition h conjugates G 2 to a symmetry group G ^ 2 of Γ γ equivalent to G 1 relative to N ( Γ γ ) . By a G 2 -equivariant isotopy, we may assume that these incompressible tori are disjoint from B 2 . Then h restricts to the identity map on B 2 . Suppose that H Diff ( S 3 , Γ γ ) realizes the above equivalence of G ^ 2 and G 1 . Then H takes Sing ( G ^ 2 ) to Sing ( G 1 ) . As a consequence of the affirmative answer to the Smith conjecture [15], Sing ( G 1 ) is either an empty set, a trivial knot, or a Hopf link whose components have different indices. Suppose that the orientation of Sing ( G 1 ) is induced from the orientation of Sing ( G ^ 2 ) by H.
Suppose that B 2 , j and B 2 , k are connected by an arc β in Sing ( G ^ 2 ) int B 2 , and that Sing ( G ^ 2 ) meets B 2 , k in an arc δ . Then B 2 , k can be modified by a G ^ 2 -equivariant deformation along β B 2 , j similar to the inverse of that mentioned above so as to contain B 2 , j . Moreover, it can be deformed along β δ B 2 , j so as to avoid B 2 , j again, as illustrated in Figure 5. Note that this operation changes the order in which the circle in Sing ( G ^ 2 ) containing β meets the balls in { B 2 , 1 , , B 2 , n } .
Let C be a component of Sing ( G 1 ) . Without loss of generality, C meets B 1 , 1 , , B 1 , r in order, and avoids B 1 , r + 1 , , B 1 , n . Since G 1 and G 2 agree on N ( Γ ) , the component H 1 ( C ) of Sing ( G ^ 2 ) meets B 2 , 1 , , B 2 , r possibly not in order. Since every permutation on the set { B 2 , 1 , , B 2 , r } is a product of transpositions realized by the above operation, we may assume that H 1 ( C ) meets B 2 , 1 , , B 2 , r in order. Apply this argument to each component of Sing ( G 1 ) . Since each B i realizes the prime factorization of E ( Γ ) , we can modify H by a G 1 -equivariant isotopy relative to N ( Γ γ ) so that we have H ( Sing ( G ^ 2 ) ) = Sing ( G 1 ) and H ( B 2 , j ) = B 1 , j for each j. Thus, H is modified so as to conjugate the action of G ^ 2 on S 3 int B 2 to the action of G 1 on S 3 int B 1 .
After this modification, H restricts to an orientation-preserving homeomorphism on B 1 . Therefore, H ( Γ B 2 ) is ambient isotopic to Γ B 1 in B 1 . Hence, H can be modified in B 1 so as to restrict to the identity map on N ( Γ ) . This completes the proof. □
Lemma 3.2.
Suppose that G 1 and G 2 are orientation-preserving finite cyclic group actions on S 2 × I such that
(1)
G 1 and G 2 do not interchange the components of S 2 × I , and
(2)
G 1 and G 2 agree on S 2 × I .
Then a rational twist along S 2 × { 1 } conjugates G 2 to a finite group action equivalent to G 1 relative to S 2 × I .
Proof. 
It is enough to consider the case where G 1 is not trivial. It follows from the remark after Theorem 8.1 of [16] that S 2 × I admits a G 1 -invariant product structure P 1 and a G 2 -invariant product structure P 2 . Since the actions of G 1 and G 2 on S 2 × { 0 } are conjugate to a rotation of S 2 (see [15]), each Fix ( G i ) consists of two I-fibers in P i . Since G 1 and G 2 agree on S 2 × I , we have Fix ( G 1 ) = Fix ( G 2 ) .
Denote by p i : S 2 × I S 2 × I / G i the projection map onto the quotient space for each i, and by S t the S 2 -fiber S 2 × { t } in P 1 . Connect the two cone points of p 1 ( S 0 ) by an arc a ¯ embedded in p 1 ( S 0 ) . Then p 1 1 ( a ¯ ) is a spatial θ n -curve consisting of two vertices on the fixed points and n > 1 edges each connecting them. Denote by A i the branched surface consisting of I-fibers in P i attaching p 1 1 ( a ¯ ) for each i. Then each p 1 ( A i S 1 ) is an arc connecting the two cone points on p 1 ( S 1 ) . Since the underlying space of p 1 ( S 1 ) is a sphere, p 1 ( A 2 S 1 ) is isotopic to p 1 ( A 1 S 1 ) relative to the cone points. Therefore, A 2 is deformed by a G 2 -equivariant isotopy relative to S 0 so that A 1 S 1 = A 2 S 1 . There are two cases depending on whether Fix ( G 1 ) and Fix ( G 2 ) are isotopic relative to the endpoints or not.
Assume that Fix ( G 1 ) and Fix ( G 2 ) are isotopic relative to the endpoints. Then P 2 is deformed by an isotopy relative to S 2 × I so as to agree with P 1 on a setwise G 1 -invariant tubular neighborhood N ( Fix ( G 1 ) ) saturated in the I-bundle structure induced from P 1 . Since each A i meets the solid torus ( S 2 × I ) int N ( Fix ( G 1 ) ) in the system of meridian disks, A 2 is moved to A 1 by an isotopy relative to ( S 2 × I ) N ( Fix ( G 1 ) ) . We may therefore assume A 1 = A 2 , and that G 1 and G 2 agree on N ( Fix ( G 1 ) ) . Then the I-bundle structures in P 1 and P 2 respectively induce the orbifold isomorphisms φ 1 : p 2 ( S 1 ) p 2 ( S 0 ) and φ 2 : p 2 ( S 0 ) p 1 ( S 1 ) such that φ 2 φ 1 is isotopic to the identity map by an isotopy relative to the cone points which setwise preserves p 2 ( A 2 S 1 ) . Then we can deform P 2 by an isotopy on p 2 ( S 2 × I ) relative to p 2 ( S 0 ) which setwise preserves p 2 ( A 2 S 1 ) so that P 1 and P 2 induce the same I -bundle structure on S 1 × I . Hence, the diffeomorphism of S 2 × I which takes P 2 to P 1 induces the equivalence of G 1 and G 2 relative to S 2 × I , as required.
Assume that Fix ( G 1 ) and Fix ( G 2 ) are not isotopic relative to the endpoints. Let h : S 1 S 1 be a lift of an orientation-preserving involution on p 1 ( S 1 ) which interchanges the cone points. Then h is a diffeomorphism isotopic to the identity map which conjugates the action of G 2 on S 1 to itself and is realized by a 1 / 2 -twist along the sphere S 1 . We may therefore assume that Fix ( G 1 ) and Fix ( G 2 ) are isotopic relative to the endpoints. Hence, the conclusion follows by the argument presented for the previous case. □
Proof of Theorem 1.1.
It is enough by Proposition 2.5 to prove the theorem in the case where Γ is splittable. Then G 1 and G 2 are cyclic groups acting on Γ freely. We may assume by Lemma 3.1 that there is a setwise G 1 -invariant and setwise G 2 -invariant system B of disjoint balls in S 3 not containing γ such that B realizes the prime factorization of E ( Γ ) , and that G 1 and G 2 agree on E ( B ) .
Suppose that B consists of balls B 1 , , B n . Each Γ B i is a non-empty, non-splittable, spatial subgraph of Γ . By applying Proposition 2.5 to the actions of the setwise stabilisers of B i in G 1 and G 2 on B i , we may assume that G 1 and G 2 agree on E ( B ) . Hence the conclusion follows by applying Lemma 3.2 to the actions of G 1 and G 2 on N ( B ) equivariantly. □
Remark 3.3.
Theorem 1.1 requires the spatial graph Γ to have no companion knot, and the symmetry groups G 1 and G 2 of Γ to act on Γ freely if Γ is splittable. These requirements are needed because of the following examples.
(1)
Suppose that Γ is a granny knot. Then Γ has two companion knots K 1 and K 2 , both of which are trefoil knots. We obtain E ( K 1 ) , E ( K 2 ) , and a 2-fold composing space by the JSJ decomposition of E ( Γ ) . Figure 6 illustrates Z 2 -symmetries G 1 and G 2 of Γ such that G 2 interchanges E ( K 1 ) and E ( K 2 ) but G 1 does not. By conjugating G 1 by a map in Diff ( S 3 ) which moves N ( Γ ) in the longitudinal direction, G 1 and G 2 are not equivalent but agree on N ( Γ ) . Moreover, any rational twists along incompressible tori in E ( Γ ) cannot change the induced symmetries of E ( K 1 ) and E ( K 2 ) , since the trefoil knot exterior is atoroidal.
(2)
Suppose that Γ is a spatial graph which splits into non-splittable spatial graphs γ 1 , γ 2 and γ 3 , as illustrated in Figure 7, where γ 1 is a spatial θ -curve. According to the choice of two edges of γ 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 Diff ( S 3 , Γ ) does not permute these edges. The Z 2 -symmetries G 1 and G 2 of Γ illustrated in Figure 7 are not equivalent, since there is no map in Diff ( S 3 , Γ ) which takes Sing ( G 1 ) to Sing ( G 2 ) and interchanges γ 2 and γ 3 . Moreover, we cannot perform rational twists along incompressible spheres and tori in E ( Γ ) to make G 2 equivalent to G 1 , since any setwise G 2 -invariant incompressible sphere in E ( Γ ) separates γ 2 and γ 3 .

Acknowledgements

The author would like to thank the referees for helpful comments which improved this paper.

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Figure 1. Conjugation by a 1 / 2 -twist along a sphere S.
Figure 1. Conjugation by a 1 / 2 -twist along a sphere S.
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Figure 2. Half twist along λ .
Figure 2. Half twist along λ .
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Figure 3. Half twist along λ .
Figure 3. Half twist along λ .
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Figure 4. Modification of p i ( B i , k ) which makes p i ( B i , k ) disjoint from p i ( B i , j ) .
Figure 4. Modification of p i ( B i , k ) which makes p i ( B i , k ) disjoint from p i ( B i , j ) .
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Figure 5. Modification of B 2 , k realizing the transposition of B 2 , j and B 2 , k .
Figure 5. Modification of B 2 , k realizing the transposition of B 2 , j and B 2 , k .
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Figure 6. Z 2 -symmetries of a spatial graph with companion knots.
Figure 6. Z 2 -symmetries of a spatial graph with companion knots.
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Figure 7. Z 2 -symmetries which are not free on a splittable spatial graph.
Figure 7. Z 2 -symmetries which are not free on a splittable spatial graph.
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