Topological symmetry groups of graphs embedded in the 3-sphere

Flapan, Erica; Naimi, Ramin; Pommersheim, James; Tamvakis, Harry

doi:10.4171/cmh/16

Cited by 24 publications

(43 citation statements)

References 18 publications

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“…It is natural to ask whether every finite group is realizable. In fact, it was shown in [3] that the alternating group A m is realizable for some graph if and only if m ≤ 5. Furthermore, in [8] it was shown that for every closed, connected, orientable, irreducible 3-manifold M , there exists an alternating group A m which is not isomorphic to the topological symmetry group of any graph embedded in M .…”

Section: Background and Terminologymentioning

confidence: 99%

“…It was shown in [3] that the set of orientation preserving topological symmetry groups of 3-connected graphs embedded in S 3 is the same up to isomorphism as the set of finite subgroups of the group of 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 (S 3 , Γ).…”

Section: Background and Terminologymentioning

confidence: 99%

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Topological Symmetry Groups of Small Complete Graphs

Chambers

Flapan

2014

Symmetry

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Section: Background and Terminologymentioning

confidence: 99%

Section: Background and Terminologymentioning

confidence: 99%

Topological Symmetry Groups of Small Complete Graphs

Chambers

Flapan

2014

Symmetry

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“…If there is a ball B in S 3 such that B ∩ Γ is an arc in the interior of e whose union with an arc in ∂B has non-trivial knot type K, then we say that B is a ball for the local knot K of e and we say that the pair (B, B ∩ Γ) has knot type K. It is shown in [4] that for any edge of an embedded 3-connected graph the local knots on that edge are well defined. If a graph is not 3-connected this is not necessarily the case.…”

Section: Unknotting Ballsmentioning

confidence: 99%

“…Flapan, Naimi, Pommersheim, and Tamvakis [4] proved that not every finite group can occur as TSG + (Γ) for some embedded graph Γ in S 3 . For example, the alternating groups A n for n > 5 cannot occur as TSG + (Γ) for any embedded graph.…”

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confidence: 99%

Spatial graphs with local knots

2011

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Abstract. It is shown that for any locally knotted edge of a 3-connected graph in S 3 , there is a ball that contains all of the local knots of that edge which is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S 3 .Schubert's 1949 result [9] that every non-trivial knot can be uniquely factored into prime knots is a fundamental result in knot theory. Hashizume [6], extended Schubert's result to links in 1958. Then in 1987, Suzuki [12] generalized Schubert's result to spatial graphs by proving that every connected graph embedded in S 3 can be split along spheres meeting the graph in 1 or 2 points to obtain a unique collection of prime embedded graphs together with some trivial graphs.Although the set of prime factors of a knot or embedded graph is unique up to equivalence, the set of splitting spheres is generally not unique up to an isotopy setwise fixing the knot or graph. For example, consider the embedding of the complete graph K 6 which is illustrated on both the left and right sides of Figure 1. The edge e = 14 contains two trefoil knots. The spheres T 1 and T 2 (illustrated on the left) are splitting spheres for these two knots. However, one of the balls bounded by F (illustrated on the right) meets e in an arc whose union with an arc in F is a single trefoil knot. Thus F is also a splitting sphere for one of the two local knots in e. However, F is not isotopic (fixing the embedded graph setwise) to either of the spheres T 1 or T 2 .By contrast, in this paper we show that for any locally knotted edge of an embedded 3-connected graph, there is a ball meeting the graph in an arc containing all of the local knots of that edge which is unique up to an isotopy fixing the graph. We call such a ball an unknotting ball for that edge. Our main theorem is the following.

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“…Unlike the complete graphs, where only some of the subgroups of SO(4) are realizable as topological symmetry groups, any finite subgroup of SO (4) can be realized as the topological symmetry group of an embedding of some K n,n [8]. So the complete bipartite graphs are a natural family of graphs to investigate in order to better understand the full range of possible topological symmetry groups.…”

Section: Introductionmentioning

confidence: 99%

Topological Symmetry Groups of Complete Bipartite Graphs

Hake¹,

Mellor²,

Pittluck³

2016

Tokyo J. Math.

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Abstract. The symmetries of complex molecular structures can be modeled by the topological symmetry group of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs [7]; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to A4, S4 or A5 [12]; in this paper we determine which complete bipartite graphs have an embedding in S 3 whose topological symmetry group is isomorphic to Zm, Dm, Zr × Zs or (Zr × Zs) ⋉ Z2.

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Topological symmetry groups of graphs embedded in the 3-sphere

Cited by 24 publications

References 18 publications

Topological Symmetry Groups of Small Complete Graphs

Topological Symmetry Groups of Small Complete Graphs

Spatial graphs with local knots

Topological Symmetry Groups of Complete Bipartite Graphs

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