Symmetry groups of flat fully augmented links and their complements

Abstract

In this paper, we prove that the (orientation-preserving) symmetry groups of $b$-prime flat fully augmented links correspond exactly with the finite subgroups of $O(3)$. We accomplish this by first developing a dictionary between automorphisms of a $3$-connected planar cubic graph associated to a flat fully augmented link $L$ and orientation-preserving symmetries of $L$. Our work also provides a simple method to explicitly construct infinite classes of distinct $b$-prime flat fully augmented links $\{L_i\}$ with $Sym^{+}(\mathbb{S}^{3} \setminus L_i) \cong Sym^{+}(\mathbb{S}^{3}, L_i) \cong G$, for any $G$ that is a finite subgroup of $O(3)$.