Symmetry groups of hyperbolic links and their complements

We explicitly construct a sequence of hyperbolic links \ L₄ₙ \ where the number of symmetries of each S³ L₄ₙ that are not induced by symmetries of the pair (S³, L₄ₙ) grows linearly with n. Specifically, [Sym(S³ L₄ₙ) : Sym(S³, L₄ₙ)] =8n → ∞ as n → ∞. For this construction, we start with a family of minimally twisted chain links, \ C₄ₙ \, where Sym(S³, C₄ₙ) and Sym(S³ C₄ₙ) coincide and grow linearly with n. We then perform a particular type of homeomorphism on S³ C₄ₙ to produce another link complement S³ L₄ₙ where we can uniformly bound |Sym(S³, L₄ₙ)| 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.