Topological Symmetries of the Heawood family
Blake Mellor, Robin Wilson
TL;DR
This work determines the realizable topological symmetry groups for every graph in the Heawood family, a class obtained from $K_7$ by $\nabla Y$ or $Y\nabla$ moves. Using tools such as Smith theory, the Finite Order Theorem, Realizability Lemma, Subgraph Lemma, Path Lemma, and the Subgroup Theorem, the authors classify automorphisms and identify, for each graph, the realizable and positively realizable groups, showing all realizable groups are positively realizable. A striking outcome is that every graph in the Heawood family is intrinsically chiral, with no embedding that is ambient isotopic to its mirror; many graphs admit explicit embeddings realizing only orientation-preserving symmetries. The results extend the catalog of spatial graphs with known ${\mathrm{TSG}}$ and illuminate how $\nabla Y$ and $Y\nabla$ moves interact with symmetry, while raising open questions about chirality preservation under these moves and potential general criteria. The findings have implications for molecular symmetries and low-dimensional topology by linking combinatorial graph structure to spatial realizability of automorphisms.
Abstract
The {\em topological symmetry group} of an embedding $Γ$ of an abstract graph $γ$ in $S^3$ is the group of automorphisms of $γ$ which can be realized by homeomorphisms of the pair $(S^3, Γ)$. These groups are motivated by questions about the symmetries of molecules in space. In this paper, we find all the groups which can be realized as topological symmetry groups for each of the graphs in the Heawood family. This is an important collection of spatial graphs, containing the only intrinsically knotted graphs with 21 or fewer edges. As a consequence, we discover that the graphs in this family are all intrinsically chiral.