Symmetry groups of hyperbolic links and their complements

We explicitly construct a sequence of hyperbolic links \ L₄₍ \ where the number of symmetries of each S^3 L₄₍ that are not induced by symmetries of the pair (S^3, L₄₍) grows linearly with n. Specifically, Sym (S^3 L₄₍): Sym (S^3, L₄₍) =8n as n. For this construction, we start with a family of minimally twisted chain links, \ C₄₍ \, where Sym (S^3, C₄₍) and Sym (S^3 C₄₍) coincide and grow linearly with n. We then perform a particular type of homeomorphism on S^3 C₄₍ to produce another link complement S^3 L₄₍ where we can uniformly bound |Sym (S^3, L₄₍) | 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.