Symmetry groups of hyperbolic flat fully augmented links and their complements

Christian Millichap
Furman

Abstract

In this talk, we will first introduce flat fully augmented links, a class of hyperbolic links whose complements admit particularly tractable geometric structures. We will then discuss how a 3-connected, planar, cubic graph called a crushtacean encodes many, and sometimes all, of the orientation-preserving symmetries of a flat fully augmented link and its complement in \(\mathbb{S}^{3}\). This combinatorial dictionary helps us show that the orientation-preserving symmetry groups of (b-prime) flat fully augmented links correspond exactly with the finite subgroups of \(O(3)\). Furthermore, given any finite subgroup \(G\) of \(O(3)\), our work provides a simple combinatorial construction to explicitly build an infinite class of distinct flat fully augmented links, \({L_i}\), where \(Sym^{+}(\mathbb{S}^{3}, L_i) \cong Sym^{+}(\mathbb{S}^{3} \setminus L_i) \cong G\). This is joint work with Rollie Trapp (CSUSB).