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

Author(s): Budney

\begin{problem} Given a link $L$ in $S^3$, let the symmetry group of $L$ be denoted $Sym(L) = \pi_0 Diff(S^3,L)$ ie: isotopy classes of diffeomorphisms of $S^3$ which preserve $L$, where the isotopies are also required to preserve $L$.

Now let $L$ be a hyperbolic link. Assume $L$ has the further `Brunnian' property that there exists a component $L_0$ of $L$ such that $L \setminus L_0$ is the unlink. Let $A_L$ be the subgroup of $Sym(L)$ consisting of diffeomorphisms of $S^3$ which preserve $L_0$ together with its orientation, and which preserve the orientation of $S^3$.

There is a representation $A_L \to \pi_0 Diff(L \setminus L_0)$ given by restricting the diffeomorphism to the $L \setminus L_0$. It's known that $A_L$ is always a cyclic group. And $\pi_0 Diff(L \setminus L_0)$ is a signed symmetric group -- the wreath product of a symmetric group with $\mathbb Z_2$.

Problem: What representations can be obtained?

% Enter your conjecture in LaTeX % You may change "conjecture" to "question" or "problem" if you prefer. \end{problem}

Keywords: knot space; symmetry

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