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

Importance: Medium ✭✭
Author(s): Budney, R
Subject: Topology
Keywords: knot space
Recomm. for undergrads: no
Posted by: rybu
on: November 7th, 2009

\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?

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An answer to this problem would give a `closed form' description of the homotopy type of the space of smooth embeddings of $S^1$ in $S^3$. This is the space of embeddings in the Whitney Topology, or $C^k$-uniform topology for any $k \geq 1$.

`Closed form' means that every component of $Emb(S^1,S^3)$ would have the description as an iterated fiber bundle over certain well-known spaces, where the fibers are inductively well-known spaces, and the monodromy would be controlled rather explicitly by this list of representations.

Peripherally related are various other realization problems for $3$-manifolds. For example, Sadayoshi Kojima proved that one can realize any finite group as the group of isometries of a hyperbolic $3$-manifold.


*[B] Budney, R. Topology of spaces of knots in dimension 3, to appear in Proc. Lond. Math. Soc.

[B2] Budney, R. A family of embedding spaces. Geometry and Topology Monographs 13 (2007).

[K] Kojima, S., Isometry transformations of hyperbolic $3$-manifolds. Topology Appl. 29 (1988), no. 3, 297--307.

* indicates original appearance(s) of problem.