Intrinsic Chirality of Graphs in 3-manifolds

TL;DR

This paper investigates the conditions under which graphs can be embedded in 3-manifolds with certain symmetry properties, revealing both restrictions and possibilities related to graph automorphisms and manifold symmetries.

Contribution

It establishes a bound on graph complexity for achiral embeddings in 3-manifolds and shows the existence of graphs with symmetric embeddings in infinitely many manifolds.

Findings

Graphs with high-genus minors cannot be achirally embedded in certain 3-manifolds.

For any graph, infinitely many 3-manifolds admit embeddings fixed by orientation-reversing involutions.

Abstract

The main result of this paper is that for every closed, connected, orientable, irreducible 3-manifold $M$, there is an integer $ n_M$ such that any abstract graph with no automorphism of order 2 which has a 3-connected minor whose genus is more than $n_M$ has no achiral embedding in $M$. By contrast, the paper also proves that for every graph $\gamma$, there are infinitely many closed, connected, orientable, irreducible 3-manifolds $M$ such that some embedding of $\gamma$ in $M$ is pointwise fixed by an orientation reversing involution of $M$.