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

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

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.

Bibliography

*[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.
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