Symmetry groups of hyperbolic links and their complements

We explicitly construct a sequence of hyperbolic links $\{L_{4n}\}$ where the number of symmetries of each ${\mathbb{S}}^3\setminus L_{4n}$ that are not induced by symmetries of the pair $({\mathbb{S}}^3,L_{4n})$ grows linearly with n. Specifically, $[Sym({\mathbb{S}}^3\setminus L_{4n}):Sym({\mathbb{S}}^3,L_{4n})]=8n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. For this construction, we start with a family of minimally twisted chain links, $\{C_{4n}\}$, where $Sym({\mathbb{S}}^3,C_{4n})$ and $Sym({\mathbb{S}}^3\setminus C_{4n})$ coincide and grow linearly with $n$. We then perform a particular type of homeomorphism on ${\mathbb{S}}^3\setminus C_{4n}$ to produce another link complement ${\mathbb{S}}^3\setminus L_{4n}$ where we can uniformly bound $\mathrm{\mid }Sym({\mathbb{S}}^3,L_{4n})\mathrm{\mid }$ using a combinatorial condition based on linking number. A more general result highlighting how to control symmetry groups of hyperbolic links is provided, which has potential for further application.