Realisation problem for the space of knots in the 3-sphere

Definitions and mathematical statement

Let $L=L_0\cup L_1\cup\cdots\cup L_m\subset S^3$ be a smooth link, with a distinguished oriented component $L_0$. Require its complement to carry a complete finite-volume hyperbolic metric, and require $L\setminus L_0$ to be an unlink, meaning that its components bound pairwise disjoint smoothly embedded disks in $S^3$. No corresponding deletion condition is imposed on the other components.

Let $A_L$ be the group of isotopy classes of orientation-preserving diffeomorphisms of $S^3$ that preserve $L$ setwise and preserve $L_0$ with its orientation. Isotopies obey the same preservation conditions. After ordering and orienting the remaining components, restriction defines a homomorphism \[ \rho_L:A_L\longrightarrow (\mathbb Z/2\mathbb Z)^m\rtimes S_m. \] The permutation records how the components move, and the signs record whether their chosen orientations reverse. The symmetric group acts on the sign factors by permuting coordinates.

Classify the homomorphisms obtained in this way as $L$ varies. The source supplies the structural fact that $A_L$ is finite cyclic; thus the target can be phrased as deciding which representations of finite cyclic groups in these signed permutation groups are realized. Representations are identified under isomorphism of their domain and conjugacy in the target, accounting for relabeling and reorientation. No faithfulness of $\rho_L$ is assumed. A realization must meet both the hyperbolicity and distinguished-component unlink conditions.

Short English statement

Which actions on the other link components can arise from symmetries of a hyperbolic link after deletion of one distinguished component leaves an unlink?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3413 (OPG-37131), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Realisation problem for the space of knots in the 3-sphere. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere.