Symmetry groups of flat fully augmented links and their complements

Paper

Abstract

In this paper, we prove that the (orientation-preserving) symmetry groups of

-prime flat fully augmented links correspond exactly with the finite subgroups of

. We accomplish this by first developing a dictionary between automorphisms of a

-connected planar cubic graph associated to a flat fully augmented link

and orientation-preserving symmetries of

. Our work also provides a simple method to explicitly construct infinite classes of distinct

-prime flat fully augmented links

with

, for any

that is a finite subgroup of