Topological Symmetry Groups for Small Complete Graphs

For each $n\leq 6$, we characterize all the groups which can occur as either the orientation preserving topological symmetry group or the topological symmetry group of some embedding of $K_n$ in $S^3$.

pieces, this molecule is achiral though it cannot be rigidly superimposed on its mirror form. A detailed discussion of the achirality of this molecule can be found in [3]. study the symmetries of arbitrary graphs embedded in 3-dimensional space, whether or not such graphs 30 represent molecules. In fact, the study of symmetries of embedded graphs can be seen as a natural 31 extension of the study of symmetries of knots which has a long history.

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Though it may seem strange from the point of view of a chemist, the study of symmetries of embedded 33 graphs as well as knots is more convenient to carry out in the 3-dimensional sphere S 3 = R 3 ∪ {∞} 34 rather than in Euclidean 3-space, R 3 . In particular, in R 3 every rigid motion is a rotation, reflection, S 3 = R 3 ∪ {∞} and extend g to a homeomorphism of S 3 simply by fixing the point at ∞. It follows 48 that the topological symmetry group of Γ in S 3 is the same as the topological symmetry group of Γ in 49 R 3 . Thus we lose no information by working with graphs in S 3 rather than graphs in R 3 .

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It was shown in [9] that the set of orientation preserving topological symmetry groups of 3-connected 51 graphs embedded in S 3 is the same up to isomorphism as the set of finite subgroups of the group of 52 orientation preserving diffeomorphisms of S 3 , Diff + (S 3 ). However, even for a 3-connected embedded graph Γ, the automorphisms in TSG(Γ) are not necessarily induced by finite order homeomorphisms of 54 (S 3 , Γ). 55 For example, consider the embedded 3-connected graph Γ illustrated in Figure  On the other hand, Flapan proved the following theorem which we will make use of later in the paper.
[4] Let ϕ be a non-trivial automorphism of a 3-connected graph γ which is 63 induced by a homeomorphism h of (S 3 , Γ) for some embedding Γ of γ in S 3 . Then for some embedding 64 Γ ′ of γ in S 3 , the automorphism ϕ is induced by a finite order homeomorphism, f of (S 3 , Γ ′ ), and f is 65 orientation reversing if and only if h is orientation reversing. 66 In the definition of the topological symmetry group, we start with a particular embedding Γ of a 67 graph γ in S 3 and then determine the subgroup of the automorphism group of γ which is induced 68 by homeomorphisms of (S 3 , Γ). However, sometimes it is more convenient to consider all possible 69 subgroups of the automorphism group of an abstract graph, and ask which of these subgroups can be the 70 topological symmetry group or orientation preserving topological symmetry group of some embedding 71 of the graph in S 3 . The following definition gives us the terminology to talk about topological symmetry 72 groups from this point of view.

Topological symmetry groups of complete graphs
For the special class of complete graphs K n embedded in S 3 , Flapan, Naimi, and Tamvakis  topological symmetry groups are possible for embeddings of a particular complete graph K n in S 3 . For 89 each n > 6, this question was answered for orientation preserving topological symmetry groups in the 90 series of papers [2,[6][7][8].

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In the current paper, we determine both the topological symmetry groups and the orientation 92 preserving topological symmetry groups for all embeddings of K n in S 3 with n ≤ 6. Another way 93 to state this is that we determine which groups are realizable and which groups are positively realizable 94 for each K n with n ≤ 6. This is the first family of graphs for which both the realizable and the positively 95 realizable groups have been determined.

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For n ≤ 3, this question is easy to answer. In particular, since K 1 is a single vertex, the only 97 realizable or positively realizable group is the trivial group. Since K 2 is a single edge, the only realizable 98 or positively realizable group is Z 2 .

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For n = 3, we know that Aut(K 3 ) ∼ = S 3 ∼ = D 3 , and hence every realizable or positively 100 realizable group for K 3 must be a subgroup of D 3 . Note that for any embedding of K 3 in S 3 , 101 the graph can be "slithered" along itself to obtain an automorphism of order 3 which is induced 102 by an orientation preserving homeomorphism. Thus the topological symmetry group and orientation 103 preserving topological symmetry group of any embedding of K 3 will contain an element of order 3. Thus 104 neither the trivial group nor Z 2 is realizable or positively realizable for K 3 . If Γ is a planar embedding 105 of K 3 in S 3 , then TSG(Γ) = TSG + (Γ) ∼ = D 3 . Recall that the trefoil knot 3 1 is chiral while the knot 106 8 17 is negative achiral and non-invertible. Thus if Γ is the knot 8 17 , then no orientation preserving 107 homeomorphism of (S 3 , Γ) inverts Γ, but there is an orientation reversing homeomorphism of (S 3 , Γ) 108 which inverts Γ. Whereas, if Γ is the knot 8 17 #3 1 , then there is no homeomorphism of (S 3 , Γ) which 109 inverts Γ.
Determining which groups are realizable and positively realizable for K 4 , K 5 , and K 6 is the main 111 point of this paper. In each case, we will first determine the positively realizable groups and then use the In addition to the Complete Graph Theorem given above, we will make use of the following results 115 in our analysis of positively realizable groups for K n with n ≥ 4. It was shown in [7] that adding a local knot to an edge of a 3-connected graph is well-defined and that 126 any homeomorphism of S 3 taking the graph to itself must take an edge with a given knot to an edge with 127 the same knot. Furthermore, any orientation preserving homeomorphism of S 3 taking the graph to itself 128 must take an edge with a given non-invertible knot to an edge with the same knot oriented in the same 129 way. Thus for n > 3, adding a distinct knot to each edge of an embedding of K n in S 3 will create an 130 embedding ∆ where TSG(∆) and TSG + (∆) are both trivial. Hence we do not include the trivial group 131 in our list of realizable and positively realizable groups for K n when n > 3. 132 Finally, observe that for n > 3, for a given embedding Γ of K n we can add identical chiral knots 133 (whose mirror image do not occur in Γ) to every edge of Γ to get an embedding Γ ′ such that TSG(Γ ′ ) = 134 TSG + (Γ). Thus every group which is positively realizable for K n is also realizable for K n . We will use 135 this observation in the rest of our analysis.

Topological Symmetry Groups of K 4 137
The following is a complete list of all the non-trivial subgroups of Aut(K 4 ) ∼ = S 4 up to isomorphism:  We will show that all of these groups are positively realizable, and hence all of the groups will also 140 be realizable. First consider the embedding Γ of K 4 illustrated in Figure 3.  Table 2.    order, then ϕ fixes at most 2 vertices. 165 We now prove the following lemma.  We summarize our results on positive realizability for K 5 in Table 3.        In order to prove D 4 is realizable for K 5 consider the embedding Γ illustrated in Figure 9. Every  We obtain a new embedding Γ ′ by replacing the invertible 4 1 knots in Figure 9 with the knot 12 427 , 216 which is positive achiral but non-invertible [14]. Since 12 427 is neither negative achiral nor invertible, no 217 homeomorphism of (S 3 , Γ ′ ) can invert 1234. Thus TSG(Γ ′ ) ∼ = Z 4 .

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Next let Γ denote the embedding of K 5 illustrated in Figure 10. Every homeomorphism of (S 3 , Γ)

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It is more difficult to show that Z 5 ⋊ Z 4 is realizable for K 5 , so we define our embedding in two steps.

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First we create an embedding Γ of K 5 on the surface of a torus T that is standardly embedded in S 3 .

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In Figure 11, we illustrate Γ on a flat torus. Let f denote a glide rotation of S 3 which rotates the torus 229 longitudinally by 4π/5 and while rotating it meridinally by 8π/5. Thus f takes Γ to itself inducing the 230 automorphism (12345).
231 Figure 11. The embedding Γ of K 5 in a torus.  S 3 about a (1, 1) curve on the torus T , followed 232 by a reflection through a sphere meeting T in two longitudes, and then a meridional rotation of T by 6π/5. In Figure 12, we illustrate the step-by-step action of g on T , showing that g takes Γ to itself 234 inducing (2431). The homeomorphisms f and g induce the automorphisms φ = (12345) and ψ = (2431) respectively.

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In order to obtain the group Z 5 ⋊ Z 4 , we now consider the embedding Γ ′ of K 5 whose projection on a 239 torus is illustrated in Figure 13. Observe that the projection of Γ ′ in every square of the grid given by Γ  Recall that g was the homeomorphism of (S 3 , Γ) obtained by rotating S 3 about a (1, 1) curve on the 243 torus T , followed by a reflection through a sphere meeting T in two longitudes, and then a meridional 244 rotation of T by 6π/5. In order to see what g does to Γ ′ , we focus on the square 1534 of Γ ′ . Figure 14   245 illustrates a rotation of the square 1534 about a diagonal, then a reflection of the square across a longitude.

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The result of these two actions takes the projection of the knot 1534 to an identical projection. Thus after 247 rotating the torus meridionally by 6π/5, we see that g takes Γ ′ to itself inducing the automorphism 248 ψ = (2431). It now follows that Z 5 ⋊ Z 4 ≤ TSG(Γ ′ ) ≤ S 5 .

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In order to prove that TSG(Γ ′ ) ∼ = Z 5 ⋊ Z 4 , we need to show TSG(Γ ′ ) ∼ = S 5 . We prove this by 250 showing that the automorphism (15) cannot be induced by a homeomorphism of (S 3 , Γ ′ ).   Thus every subgroup of Aut(K 5 ) is realizable for K 5 . Table 4 summarizes our results for TSG(K 5 ).      The following is a complete list of all the subgroups of Aut(K 6 ) ∼ = S 6 : and independently verified using the program GAP).    Observe that every homeomorphism of (S 3 , Γ) takes the pair of triangles 123 ∪ 456 to itself, since 294 The subgroup f, g, ψ is isomorphic to D 3 × Z 3 because ψ commutes with f and gψ = ψg −1 . We 295 add the non-invertible knot 8 17 to every edge of the triangles 123 and 456 to obtain an embedding Γ 1 .

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Now the automorphism φ = (45)(12) cannot be induced by an orientation preserving homeomorphism 297 of (S 3 , Γ 1 ). However, f , g, and ψ are still induced by orientation preserving homeomorphisms. Thus with Γ in Figure 19, we place 5 2 knots on the edges of the triangle 123 so that ψ is no longer induced.

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Thus creating and embedding Finally f, g is isomorphic to Z 3 × Z 3 . If we place equivalent non-invertible knots on each edge of 304 the triangle 123 and a another set (distinct from the first set) of equivalent non-invertible knots on each 305 edge of 456 we obtain an embedding Γ 3 with TSG + (Γ 3 ) ∼ = Z 3 ×Z 3 since Z 3 ×Z 3 is a maximal subgroup 306 of (Z 3 × Z 3 ) ⋊ Z 2 .      For the next few groups we will use the following lemma. 335 4-Cycle Theorem.
[5] For any embedding Γ of K 6 in S 3 , and any labelling of the vertices of K 6 by the 336 numbers 1 through 6, there is no homeomorphism of (S 3 , Γ) which induces the automorphism (1234).

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Consider the subgroup Z 5 ⋊ Z 4 ≤ Aut(K 6 ). The presentation of Z 5 ⋊ Z 4 as a subgroup of S 6 gives 338 the relation x −1 yx = y 2 for some elements x, y ∈ Z 5 ⋊ Z 4 of orders 4 and 5 respectively. Suppose that 339 for some embedding Γ of K 6 , we have TSG(Γ) ∼ = Z 5 ⋊ Z 4 . Without loss of generality, we can assume  This rules out all of the groups S 4 × Z 2 , D 4 × Z 2 and Z 4 × Z 2 as possible topological symmetry groups 352 for embeddings of K 6 in S 3 .

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For the group Z 2 × Z 2 × Z 2 we will use the following result. Since h, f , and g are homeomorphisms of (S 3 , Γ) the links in a given orbit all have the same (mod 364 2) linking number. Since each of these orbits has an even number of pair of triangles, this contradicts 365 Conway Gordon. Thus Z 2 × Z 2 × Z 2 ∼ = TSG(Γ). Hence Z 2 × Z 2 × Z 2 is not realizable for K 6 366 Table 6 summarizes our realizability results for K 6 . Recall that for n = 4 and n = 5 every subgroup 367 of S n is realizable for K n . However, as we see from Table 6, this is not true for n = 6. she was a long term visitor in the fall of 2013.