On defining sequences for Cantor sets

Each Cantor set can have many essentially different defining sequences and there is no canonical way to choose one. The natural approach of reducing the number of handles does not work if the complement of a given Cantor set is too nice (for example, simply connected). shown that there are no “unnecessary handles” in the defining sequence of a given Cantor if and only if the Cantor set satisfies some weakened 1-ULC property.