Intrinsic chirality of graphs in 3-manifolds

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 if $\gamma$ is a graph with no involution and a 3-connected minor $\lambda$ with $\mathrm{genus}(\lambda) \gt n_M$, then every embedding of $\gamma$ in $M$ is chiral. 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 $M$.

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The first author was supported in part by NSF Grant DMS-1607744.

Received 14 December 2015

Published 29 March 2019