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Topological symmetry groups were originally introduced to study the symmetries of non-rigid molecules, but have since been used to study the symmetries of any graph embedded in ℝ^{3}. In this paper, we determine for each complete graph _{n}^{3}.

Molecular symmetries are important in many areas of chemistry. Symmetry is used in interpreting results in crystallography, spectroscopy, and quantum chemistry, as well as in analyzing the electron structure of a molecule. Symmetry is also used in designing new pharmaceutical products. But what is meant by a “symmetry” depends on the rigidity of the molecule in question.

For rigid molecules, the group of rotations, reflections, and combinations of rotations and reflections, is an effective way of representing molecular symmetries. This group is known as the ^{3}. However, some molecules can rotate around particular bonds, and large molecules can even be somewhat flexible. For example, supramolecular structures constructed through self-assembly may be somewhat conformationally flexible. Even relatively small molecules may contain rigid molecular subparts that rotate on hinges around particular bonds. For example, the left and right sides of the biphenyl derivative illustrated in

In general, the amount of rigidity of a given molecule depends on its chemistry not just its geometry. Thus a purely mathematical definition of molecular symmetries that accurately reflects the behavior of all molecules is impossible. However, for non-rigid molecules, a topological approach to classifying symmetries including achirality can add important information beyond what is obtained from the point group. Such an approach could be useful to the study of supramolecular chirality, since structures constructed through self-assembly may be large and somewhat flexible or contain subparts that can rotate around covalent or non-convalent bonds.

The

In this paper, we determine both the topological symmetry groups and the orientation preserving topological symmetry groups of structures whose underlying form is that of a complete graph with no more than six vertices. A _{n}_{3}, _{4}, and _{5}. The class of complete graphs is an interesting class to consider because the automorphism group of _{n}_{n}_{n}_{4}. If such a cluster contains a central atom which is bonded to the four corners of the tetrahedron, then the structure has the form of the complete graph _{5} (as illustrated on the right in

Though it may seem strange from the point of view of a chemist, the study of symmetries of embedded graphs is more convenient to carry out in the 3-dimensional sphere ^{3} = ℝ^{3} ∪ {∞} rather than in Euclidean 3-space, ℝ^{3}. In particular, in ℝ^{3} every rigid motion is a rotation, reflection, translation, or a combination of these operations. Whereas, in ^{3} glide rotations provide an additional type of rigid motion. While a topological approach to the study of symmetries does not require us to focus on rigid motions, for the purpose of illustration it is preferable to display rigid motions rather than isotopies whenever possible. Thus throughout the paper we work in ^{3} rather than in ℝ^{3}.

The ^{3}, Γ). The _{+}(Γ), is the subgroup of Aut(Γ) induced by orientation preserving homeomorphisms of (^{3}, Γ).

It should be noted that for any homeomorphism ^{3}, Γ), there is a homeomorphism ^{3}, Γ) which fixes a point ^{3}, Γ). On the other hand if we start with an embedded graph Γ in ℝ^{3} and a homeomorphism ^{3}, Γ), we can consider ^{3} = ℝ^{3} ∪ {^{3} simply by fixing the point at ∞. It follows that the topological symmetry group of Γ in ^{3} is the same as the topological symmetry group of ^{3}. Thus we lose no information by working with graphs in ^{3} rather than graphs in ℝ^{3}.

It was shown in [^{3} is the same up to isomorphism as the set of finite subgroups of the group of orientation preserving diffeomorphisms of ^{3}, Diff_{+}(^{3}). However, even for a 3-connected embedded graph Γ, the automorphisms in TSG(Γ) are not necessarily induced by finite order homeomorphisms of (^{3}, Γ).

For example, consider the embedded 3-connected graph Γ illustrated in ^{3} because there is no order three homeomorphism of ^{3} taking a figure eight knot to itself [^{3} [

On the other hand, Flapan proved the following theorem which we will make use of later in the paper.

[^{3}, Γ) ^{3}. ^{3}, ^{3}, Γ),

In the definition of the topological symmetry group, we start with a particular embedding Γ of a graph ^{3} and then determine the subgroup of the automorphism group of ^{3},Γ). However, sometimes it is more convenient to consider all possible subgroups of the automorphism group of an abstract graph, and ask which of these subgroups can be the topological symmetry group or orientation preserving topological symmetry group of some embedding of the graph in ^{3}. The following definition gives us the terminology to talk about topological symmetry groups from this point of view.

^{3} ^{3}, Γ). ^{3} _{+}(Γ) =

It is natural to ask whether every finite group is realizable. In fact, it was shown in [_{m}_{m}

For the special class of complete graphs _{n}^{3}, Flapan, Naimi, and Tamvakis obtained the following result.

[_{+}(Γ) ^{3} _{m}_{m}for some odd m

This left open the question of what topological symmetry groups and orientation preserving topological symmetry groups are possible for embeddings of a particular complete graph _{n}^{3}. For each _{n}_{n}

In the current paper, we determine which groups are realizable and which groups are positively realizable for each _{n}

For _{1} is a single vertex, the only realizable or positively realizable group is the trivial group. Since _{2} is a single edge, the only realizable or positively realizable group is ℤ_{2}.

For _{3}) ≅ _{3} ≅ D_{3}, and hence every realizable or positively realizable group for _{3} must be a subgroup of _{3}. Note that for any embedding of _{3} in ^{3}, the graph can be “slithered” along itself to obtain an automorphism of order 3 which is induced by an orientation preserving homeomorphism. Thus the topological symmetry group and orientation preserving topological symmetry group of any embedding of _{3} will contain an element of order 3. Thus neither the trivial group nor Z_{2} is realizable or positively realizable for _{3}_{3}^{3}, then TSG(Γ) = TSG_{+}(Γ) ≅ D_{3}. Recall that the trefoil knot 31 is chiral while the knot 81_{7} is negative achiral and non-invertible. Thus if Γ is the knot 8_{17}, then no orientation preserving homeomorphism of (^{3}, Γ) inverts Γ, but there is an orientation reversing homeomorphism of (^{3}, Γ) which inverts Γ. Whereas, if Γ is the knot 8_{17}#3_{1}, then there is no homeomorphism of (^{3}, Γ) which inverts Γ. _{3}.

Determining which groups are realizable and positively realizable for _{4}, _{5}, and _{6} is the main point of this paper. In each case, we will first determine the positively realizable groups and then use the fact that either TSG_{+}(Γ) = TSG(Γ) or TSG_{+}(Γ) is a normal subgroup of TSG(Γ) of index 2 to help us determine the realizable groups.

In addition to the Complete Graph Theorem given above, we will make use of the following results in our analysis of positively realizable groups for _{n}

[_{m} with m ≥ 4 has an embedding Γ in ^{3} such that TSG_{+}(Γ) ≅ A_{4} if and only ifm ≡ 0, 1, 4, 5, 8 (mod 12).

[_{m} with m ≥ 4 has an embedding Γ in ^{3} such that TSG_{+}(Γ) ≅ A_{5} if and only ifm ≡ 0, 1, 5, 20 (mod 60).

[_{m} with m ≥ 4 has an embedding Γ in ^{3} such that TSG_{+}(Γ) ≅ S_{4} if and only if m ≡ 0, 4, 8, 12, 20 (mod 24).

[^{3} with an edge that is not poinwise fixed by any non-trivial element of TSG_{+}(Γ). Then every subgroup of TSG_{+}(Γ) is positively realizable for γ.

It was shown in [^{3} taking the graph to itself must take an edge with a given knot to an edge with the same knot. Furthermore, any orientation preserving homeomorphism of ^{3} taking the graph to itself must take an edge with a given non-invertible knot to an edge with the same knot oriented in the same way Thus for _{n}^{3} will create an embedding Δ where TSG(Δ) and TSG_{+}(Δ) are both trivial. Hence we do not include the trivial group in our list of realizable and positively realizable groups for _{n}

Finally, observe that for _{n}_{+}(Γ). Thus every group which is positively realizable for _{n}_{n}

The following is a complete list of all the non-trivial subgroups of Aut(_{4}) ≅ S_{4} up to isomorphism: S_{4}, _{4}, D_{4}, D_{3}, D_{2}, ℤ_{4}, ℤ_{3}, ℤ_{2}.

We will show that all of these groups are positively realizable, and hence all of the groups will also be realizable. First consider the embedding Γ of _{4} illustrated in ^{3}, Γ). Hence TSG_{+}(Γ) is a subgroup of D_{4}. In order to obtain the automorphism (1234), we rotate the square
_{+}(Γ) ≅ D_{4}. Furthermore, since the edge
_{+}(Γ), by the Subgroup Theorem the groups ℤ_{4}, D_{2} and ℤ_{2} are each positively realizable for _{4}.

Next, consider the embedding, Γ of _{4} illustrated in ^{3}, Γ) fix vertex 4. Hence TSG_{+}(Γ) is a subgroup of _{3}. The automorphism (123) is induced by a rotation, and the automorphism (12) is induced by turning the figure upside down and then pushing vertex 4 back up through the centre of 123. Thus TSG_{+}(Γ) = D_{3}. Since the edge 12 is not pointwise fixed by any non-trivial element of TSG_{+}(Γ), by the Subgroup Theorem, the group Z_{3} is also positively realizable for _{4}.

Thus every subgroup of Aut(_{4}) is positively realizable. Now by adding appropriate equivalent chiral knots to each edge, all subgroups of Aut(_{4}) are also realizable. We summarize our results for _{4}

The following is a complete list of all the non-trivial subgroups of Aut(_{5}) ≅ S_{5}:

S_{5}, _{5}_{4}, _{4}, ℤ_{5} × ℤ_{4}, D_{6}, D_{5}, D_{4}, D_{3}, D_{2}, ℤ_{6}, ℤ_{5}, ℤ_{4}, ℤ_{3}, ℤ_{2} (see [

The lemma below follows immediately from the Finite Order Theorem [

_{n} which is induced by a homeomorphism h of^{3}, Γ) _{n} in S^{3}.

We now prove the following lemma.

_{n} in S^{3} _{+}(Γ)

By the Finite Order Theorem, _{n}^{3}, Γ′). Suppose that ^{1}. Let ^{r}_{1}_{1}) … _{q}b_{q})^{r}_{i}b^{r}^{r}_{i}b_{i}_{1} _{1}) … _{q}b_{q})

By Lemma 2, there is no embedding of _{5} in ^{3} such that TSG_{+}(Γ) contains an element of order 4 or of order 6. It follows that TSG_{+}(Γ) cannot be D_{6}, ℤ_{6}, D_{4} or ℤ_{4}.

Consider the embedding Γ of _{5} illustrated in _{+}(Γ) ≤ _{5}. The automorphism (12345) is induced by rotating Γ, and (25)(34) is induced by turning the graph over. Hence TSG_{+}(Γ) = 〈(12345;, (25)(34)〉 ≅ D_{5}. Since the edge
_{+}(Γ), by the Subgroup Theorem the groups ℤ_{5} and ℤ_{2} are also positively realizable for _{5}.

Next consider the embedding Γ of _{5} illustrated in _{5}. Thus TSG_{+}(Γ) ≤ D_{3}. The automorphism (123) is induced by a rotation. Also the automorphism (45)(12) is induced by pulling vertex 4 down through the centre of triangle
_{+}(Γ) = 〈(123;, (45)(12)〉 ≅ D_{3}. Since the edge
_{+}(Γ), by the Subgroup Theorem, the group ℤ_{3} is positively realizable for _{5}.

Lastly, consider the embedding Γ of _{5} illustrated in _{+}(Γ) ≤ D_{4}. The automorphism (13)(24) is induced by rotating the square by 180°. By turning over the figure we obtain (12)(34). By Lemma 2, TSG_{+}(Γ) cannot contain an element of order 4. Thus TSG_{+}(Γ) = 〈(13)(24;, (12)(34)〉 ≅ D_{2}.

We summarize our results on positive realizability for _{5} in

Again by adding appropriate equivalent chiral knots to each edge, all of the positively realizable groups for _{5} are also realizable. Thus we only need to determine realizability for the groups S_{5}, S_{4}, ℤ_{5} ⋊ ℤ_{4}, D_{6}, D_{4}, ℤ_{6},andℤ_{4}.

Let Γ be the embedding of _{5} illustrated in _{5}.

We create a new embedding Γ from _{1}, to all edges containing vertex 5. Now every homeomorphism of (^{3}, Γ) fixes vertex 5, yet all transpositions fixing vertex 5 are still possible. Thus TSG(Γ′) ≅ S_{4}.

In order to prove D_{4} is realizable for _{5} consider the embedding Γ illustrated in ^{3}, Γ) takes
_{4}. The automorphism (1234) is induced by rotating the graph by 90° about a vertical line through vertex 5, then reflecting in the plane containing the vertices 1, 2, 3, 4, and finally isotoping the knots into position. Furthermore, reflecting in the plane containing
_{4}.

We obtain a new embedding Γ by replacing the invertible 41 knots in _{427}, which is positive achiral but non-invertible [_{427} is neither negative achiral nor invertible, no homeomorphism of (^{3}, Γ) can invert
_{4}.

Next let Γ denote the embedding of _{5}^{3}, Γ) takes
_{6}. The 3-cycle (123) is induced by a rotation. Each transposition involving only vertices 1, 2, and 3 is induced by a reflection in the plane containing
_{6}, generated by (123), (23), and (45).

We obtain a new embedding Γ′ by replacing the 4_{1} knots in _{427} knots. Then the triangle
_{6}, generated by (123) and (45).

It is more difficult to show that ℤ_{5} × ℤ_{4} is realizable for _{5}, so we define our embedding in two steps. First we create an embedding Γ of _{5} on a torus ^{3}. In ^{3} which rotates the torus longitudinally by 4

Let ^{3} about a (1, 1) curve on the torus

The homeomorphisms ^{5} ^{4} = 1 and ^{2}. Thus 〈_{5} ⋊ ℤ_{4} ≤ TSG(Γ) ≤ S_{5}. Note however that the embedding in _{5}_{5}.

In order to obtain the group ℤ_{5} ⋊ ℤ_{4}, we now consider the embedding Γ of _{5} whose projection on a torus is illustrated in

Recall that ^{3}, Γ) obtained by rotating ^{3} about a (1, 1) curve on the torus _{5} ⋊ ℤ_{4} ≤ TSG(Γ′) < _{5}.

In order to prove that TSG(Γ′) ≅ ℤ_{5} ⋊ ℤ_{4}, we need to show TSG(Γ′) ≇ S_{5}. We prove this by showing that the automorphism (15) cannot be induced by a homeomorphism of (^{3}, Γ′).

From _{1}#4_{1}#4_{1}#4_{1}. In order to see what would happen to this knot if the transposition (15) were induced by a homeomorphism of (^{3}, Γ), we consider the square
_{1}#4_{1}#4_{1}#4_{1}. It follows that the automorphism (15) cannot be induced by a homeomorphism of (^{3}, Γ). Hence TSG(Γ′) ≇ _{5}. However, the only subgroup of S_{5} that contains ℤ_{5} ⋊ ℤ_{4} and is not _{5}_{5} ⋊ ℤ_{4}. Thus in fact TSG(Γ′) ≅ ℤ_{5} ⋊ ℤ_{4}.

Thus every subgroup of Aut(_{5}) is realizable for _{5}. _{5}).

The following is a complete list of all the subgroups of Aut(_{6}) ≅ S_{6}: S_{6}, _{6}, s_{5}, _{5}, _{2}[_{3}_{4} × ℤ_{2}, _{4} × ℤ_{2}, S_{4}, _{4}, ℤ_{5} ⋊ ℤ_{4}, D_{3} × D_{3}, (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4}, (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{2}, D_{3} × ℤ_{3}, ℤ_{3} × ℤ_{3}, D_{6}, D_{5}, D_{4}, D_{4} × ℤ_{2}, D_{3}, D_{2}, ℤ_{6}, ℤ_{5}, ℤ_{4}, ℤ_{4} × ℤ_{2}, ℤ_{3}, ℤ_{2}, ℤ_{2} × ℤ_{2} × ℤ_{2} (see [

Consider the embedding Γ of _{6} illustrated in _{+}(Γ) < _{6}. The automorphism (123456) is induced by a glide rotation that cyclically permutes the Z-paths. Consider the homeomorphism obtained by rotating Γ by 180° about the line through vertices 2 and 5 and then pulling the edges
_{+}(Γ) = 〈(123456;, (13)(46)〉 ≅ D_{6}. Finally, since the edge
_{+}(Γ), by the Subgroup Theorem the groups ℤ_{6}, D_{3}, ℤ_{3}, D_{2} and ℤ_{2} are positively realizable for _{6}.

Consider the embedding, Γ of _{6} illustrated in ^{3}, Γ). Hence TSG_{+}(Γ) ≅ D_{5}. Also since 15 is not pointwise fixed under any homeomorphism, by the Subgroup Theorem, ℤ_{5} is positively realizable for _{6}.

Next consider the embedding, Γ of _{6} illustrated in

Observe that every homeomorphism of (^{3}, Γ) takes the pair of triangles
_{2}[_{3}] [_{+}(Γ) ≤ _{2}[_{3}]. Note that the transpositions (12) and (45) are each induced by a reflection followed by an isotopy. Thus TSG(Γ) ≅ _{2}[_{3}], since (123)(456), (123)(465), (12) and (14)(25)(36) generate _{2}[S_{3}]_{+}(Γ) ≇ _{2}[_{3}]. Thus TSG_{+}(Γ) must be an index 2 subgroup of _{2}[_{3}] containing ^{-}^{1}, while ^{-1}. Thus _{2}[_{3}] ≥ TSG_{+}(Γ) ≥ 〈_{3} × D_{3}. It follows that TSG+(Γ) ≅ D_{3} × D_{3}.

The subgroup 〈_{3} × ℤ_{3} because ^{-1}. We add the non-invertible knot 8_{17} to every edge of the triangles
_{1}. Now the automorphism ^{3}, Γ_{1}). However, _{+}(Γ_{1}) ≅ D_{3} × ℤ_{3} since D_{3} × ℤ_{3} is a maximal subgroup of D_{3} x D_{3}.

Also 〈_{3} × ℤ_{3}) ⋊ ℤ_{2} because ^{-}^{1} and ^{-1}. Again starting with Γ in _{2} knots on the edges of the triangle 123 so that _{2} with TSG_{+}(Γ_{2}) ≅ (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{2} since (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{2} is a maximal subgroup of D_{3} × D_{3}.

Finally 〈_{3} × ℤ_{3}. If we place equivalent non-invertible knots on each edge of the triangle
_{3} with TSG_{+}(Γ_{3}) ≅ ℤ_{3} × ℤ_{3} since ℤ_{3} × ℤ_{3} is a maximal subgroup of (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{2}.

We summarize our results on positively realizability for _{6} in

By adding appropriate equivalent chiral knots to each edge, every group which is positively realizable for _{6} is also realizable for _{6}. Thus we only need to determine realizability for the groups S_{6}, _{6}, S_{5}, _{5}, S_{4} × ℤ_{2}, _{4} × ℤ_{2}, S_{4}, _{4}, ℤ_{5} ⋊ ℤ_{4}, (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4}, D_{4}, D_{4} × ℤ_{2}, ℤ_{4}, ℤ_{4} × ℤ_{2}, and ℤ_{2} × ℤ_{2} × ℤ_{2}. Note that in _{2}[_{3}] is realizable for _{6}.

Let Γ_{4} be the embedding of _{6} illustrated in _{4},^{3}). Also if we reflect in the plane containing vertices 4, 5, 6, and 1 then all the trefoils switch from left-handed to right-handed and vice versa. If we then interchange the complementary solid tori which have the triangles as cores followed by an isotopy, we obtain an orientation reversing homeomorphism that induces the order 4 automorphism (14)(25)(36)(23) = (14)(2536). Now 〈(14)(2536;, (123)(456), (123)(465)〉 ≅ (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4}.

We see as follows that TSG(Γ_{4}) cannot be larger than (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4}. Suppose that the automorphism (12) is induced by a homeomorphism _{4}) ≇ _{2}[_{3}]. Note that the only proper subgroup of _{2}[_{3}] containing (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4} is (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4}. Thus TSG(Γ_{4}) ≅ (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4}.

Now let Γ be the embedding of _{6}^{3},Γ). Since every homeomorphism of (^{3},Γ) takes
_{4}. The automorphism (1234)(56) is induced by a rotation followed by a reflection and an isotopy. In addition the automorphism (14)(23)(56) is induced by turning the figure upside down. Thus TSG(Γ) ≅ D_{4} generated by the automorphisms (1234)(56) and (14)(23)(56).

Now let Γ be obtained from _{1} with the non-invertible and positively achiral knot 12_{427}. Then the square
_{4}.

For the next few groups we will use the following lemma.

[_{6} in ^{3}, and any labelling of the vertices of K_{6} by the numbers 1 through 6, there is no homeomorphism of (^{3},Γ) which induces the automorphism (1234).

Consider the subgroup ℤ_{5} ⋊ ℤ_{4} ≤ Aut(_{6}). The presentation of ℤ_{5} ⋊ ℤ_{4} as a subgroup of _{6}^{-1}^{2} for some elements _{5} ⋊ ℤ_{4} of orders 4 and 5 respectively. Suppose that for some embedding Γ of _{6}_{5} ⋊ ℤ_{4}. Without loss of generality, we can assume that ^{-1}^{2}. By the 4-Cycle Theorem, any order 4 element of TSG(Γ) must be of the form _{6}) of the form _{6} in ^{3} such that TSG(Γ) ≅ ℤ_{5} ⋊ ℤ_{4}.

Now consider the subgroup ℤ_{4} × ℤ_{2} ≤ Aut(_{6}). By the 4-Cycle Theorem, without loss of generality we may assume that if TSG(Γ) contains an element of order 4 for some embedding Γ of _{6}, then TSG(Γ) contains the element (1234)(56). Computation shows that the only transposition in Aut(_{6}) that commutes with (1234)(56) is (56), which cannot be an element of TSG(Γ) since this would imply that (1234) is an element of TSG(Γ). Furthermore the only order 2 element of Aut(_{6}) that commutes with (1234)(56) and is not a transposition is (13)(24), which is already in the group generated by (1234)(56). Thus there is no embedding Γ of _{6}^{3} such that TSG(Γ) contains the group ℤ_{4} × ℤ_{2}. 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 for embeddings of _{6} in ^{3}.

For the group ℤ_{2} × ℤ_{2} × ℤ_{2} we will use the following result.

[_{6} ^{3},

Now suppose that for some embedding Γ of _{6} in ^{3} we have TSG(Γ) ≅ ℤ_{2} × ℤ_{2} × ℤ_{2}. It can be shown that the subgroup ℤ_{2} × ℤ_{2} × ℤ_{2} ≤ Aut(_{6}) contains three disjoint transpositions. Without loss of generality we can assume that TSG(Γ) contains (13), (24), and (56), which are induced by homeomorphisms ^{3}, Γ) respectively. Since any three vertices of Γ determine a pair of disjoint triangles, we can use a triple of vertices to represent a pair of disjoint triangles. For example, we use the triple 123 to denote the pair of triangles
_{6} under the group 〈(13;, (24), (56)〉 are:
^{3}, Γ) the links in a given orbit all have the same (mod 2) linking number. Since each of these orbits has an even number of pair of triangles, this contradicts Conway Gordon. Thus ℤ_{2} × ℤ_{2} × ℤ_{2} ≇ TSG(Γ). Hence ℤ_{2} × ℤ_{2} × ℤ_{2} is not realizable for _{6}

_{6}. Recall that for _{n}_{n}

We have classified all groups which can occur as the topological symmetry group or orientation preserving topological symmetry group of an embedded complete graph with no more than six vertices. Our results show that a number of groups can occur as a topological symmetry group but not as an orientation preserving topological symmetry group for a particular _{n}

The topological symmetry group includes all of the symmetries induced by the point group together with any symmetries that occur as the result of any flexibility or rotation of subparts of a structure around specific bonds. Thus the topological symmetry group gives us more information about the symmetries and possible achirality of supramolecular structures than could be obtained from the point group. Since complete graphs with no more than six vertices may occur as supramolecular clusters, these results could be of interest in the future study of supramolecular chirality.

The first author would like to thank Claremont Graduate University for its support while he pursued the study of Topological Symmetry Groups for his Ph.D Thesis. The second author would like to thank the Institute for Mathematics and its Applications at the University of Minnesota for its hospitality while she was a long term visitor in the fall of 2013.

The authors declare no conflict of interest.

_{4}

_{r}

_{+3}

_{n}

_{5}

_{6}

Because of its rotating subparts, this molecule is achiral.

The graphs _{3}, _{4}, and _{5}.

The topological symmetry group of this embedded graph is not induced by a finite group of homeomorphismsof ^{3}.

TSG_{+}(Γ) ≅ D_{4}.

TSG_{+}(Γ) ≅ D_{3}.

TSG_{+}(Γ) ≅ D_{5}.

TSG_{+}(Γ) ≅ D_{3}.

TSG_{+}(Γ) ≅ D_{2}.

TSG(Γ) ≅ S_{5}.

TSG(Γ) ≅ D_{4}.

TSG(Γ) ≅ D_{6}.

The embedding Γ of _{5} in a torus.

The action of

Projection of Γ′ on the torus.

Effect of

The knot

A projection of

TSG_{+}(Γ) ≅ D_{6}.

TSG_{+}(Γ) ≅ D_{5}.

TSG_{+}(Γ) ≅ D_{3}×D_{3}.

TSG(Γ) ≅ D_{4}.

Realizable and positively realizable groups for _{3}.

_{+}(Γ) | ||
---|---|---|

Planar | D_{3} |
D_{3} |

8_{17} |
D_{3} |
ℤ_{3} |

8_{17} #3_{1} |
ℤ_{3} |
ℤ_{3} |

Non-trivial realizable and positively realizable groups for _{4}.

S_{4} |
Yes | By S_{4} Theorem |

_{4} |
Yes | By _{4} Theorem |

D_{4} |
Yes | By |

D_{3} |
Yes | By |

D_{2} |
Yes | By Subgroup Theorem |

ℤ_{4} |
Yes | By Subgroup Theorem |

ℤ_{3} |
Yes | By Subgroup Theorem |

ℤ_{2} |
Yes | By Subgroup Theorem |

Non-trivial positively realizable groups for _{5}.

S_{5} |
No | By Complete Graph Theorem |

_{5} |
Yes | By _{5} Theorem |

ℤ_{5} ⋊ ℤ_{4} |
No | By Complete Graph Theorem |

S_{4} |
No | By S_{4} Theorem |

_{4} |
Yes | By _{4} Theorem |

D_{6} |
No | By Lemma 2 |

D_{5} |
Yes | By |

D_{4} |
No | By Lemma 2 |

D_{3} |
Yes | By |

D_{2} |
Yes | By |

ℤ_{6} |
No | By Lemma 2 |

ℤ_{5} |
Yes | By Subgroup Theorem |

ℤ_{4} |
No | By Lemma 2 |

ℤ_{3} |
Yes | By Subgroup Theorem |

ℤ_{2} |
Yes | By Subgroup Theorem |

Non-trivial realizable groups for _{5}.

S_{5} |
Yes | By |

_{5} |
Yes | Positively realizable |

S_{4} |
Yes | By modifying |

_{4} |
Yes | Positively realizable |

D_{6} |
Yes | By |

D_{5} |
Yes | Positively realizable |

D_{4} |
Yes | By |

D_{3} |
Yes | Positively realizable |

D_{2} |
Yes | Positively realizable |

ℤ_{6} |
Yes | By modifying |

ℤ_{5}⋊ℤ_{4} |
Yes | By |

ℤ_{5} |
Yes | Positively realizable |

ℤ_{4} |
Yes | By modifying |

ℤ_{3} |
Yes | Positively realizable |

ℤ_{2} |
Yes | Positively realizable |

Non-trivial positively realizable groups for _{6}.

_{5} |
No | By _{5} Theorem |

S_{4} |
No | By S_{4} Theorem |

_{4} |
No | By _{4} Theorem |

D_{6} |
Yes | By |

D_{5} |
Yes | By |

D_{4} |
No | By Lemma 2 |

D_{3}×D_{3} |
Yes | By |

D3×ℤ3 | Yes | By modifying |

D_{3} |
Yes | By Subgroup Theorem |

D_{2} |
Yes | By Subgroup Theorem |

ℤ_{6} |
Yes | By Subgroup Theorem |

ℤ_{5} |
Yes | By Subgroup Theorem |

ℤ_{4} |
No | By Lemma 2 |

(ℤ_{3} × ℤ_{3}) x ℤ_{2} |
Yes | By modifying |

ℤ_{3} ×ℤ_{3} |
Yes | By modifying |

ℤ_{3} |
Yes | By Subgroup Theorem |

ℤ_{2} |
Yes | By Subgroup Theorem |

S_{6},_{6},S_{5},_{2}[_{3}],S_{4}×ℤ_{2},_{4}×ℤ_{2} |
No | By Complete Graph Theorem |

ℤ_{5} ⋊ ℤ_{4}, (ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4}, D_{4} × ℤ_{2} |
No | By Complete Graph Theorem |

ℤ_{4} ×ℤ_{2}, ℤ_{2} _{2} _{2} |
No | By Complete Graph Theorem |

Non-trivial realizable groups for _{6}.

S_{6} |
No | TSG_{+}(_{6}) cannot be S_{6} or _{6} |

_{6} |
No | TSG_{+}(_{6}) cannot be _{6} |

S_{5} |
No | TSG_{+}(_{6}) cannot be S_{5} or _{5} |

_{5} |
No | TSG_{+}(_{6}) cannot be _{5} |

S_{4} × ℤ_{2} |
No | TSG+(_{6}) cannot be S_{4} × ℤ_{2} or S_{4} |

S_{4} |
No | TSG_{+}(_{6}) cannot be S_{4} or _{4} |

_{4} × ℤ_{2} |
No | TSG_{+}(_{6})cannotbe_{4} × ℤ_{2}or_{4} |

_{4} |
No | TSG_{+}(_{6}) cannot be _{4} |

D_{6} |
Yes | Positively realizable |

D_{5} |
Yes | Positively realizable |

D_{4}×ℤ_{2} |
No | TSG+(_{6}) cannot be D_{4} × ℤ_{2}, D_{4}, ℤ_{4} × ℤ_{2}, ℤ_{2} × ℤ_{2} × ℤ_{2} |

D_{4} |
Yes | By |

_{2}[_{3}] |
Yes | By |

D_{3} × D_{3} |
Yes | Positively realizable |

D_{3} × ℤ_{3} |
Yes | Positively realizable |

D_{3} |
Yes | Positively realizable |

D_{2} |
Yes | Positively realizable |

ℤ_{6} |
Yes | Positively realizable |

ℤ_{5}×ℤ_{4} |
No | By 4-Cycle Theorem |

ℤ_{5} |
Yes | Positively realizable |

ℤ_{4}×ℤ_{2} |
No | By 4-Cycle Theorem |

ℤ_{4} |
Yes | By modifying |

(ℤ_{3} × ℤ_{3}) ⋊ ℤ_{4} |
Yes | By modifying |

(ℤ_{3} × ℤ_{3}) ⋊ ℤ_{2} |
Yes | Positively realizable |

ℤ_{3}×ℤ_{3} |
Yes | Positively realizable |

ℤ_{3} |
Yes | Positively realizable |

ℤ_{2} x ℤ_{2} × ℤ_{2} |
No | By Conway Gordon Theorem |

ℤ_{2} |
Yes | Positively realizable |