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knot space
Realisation problem for the space of knots in the 3-sphere ★★
Author(s): Budney
Problem Given a link
in
, let the symmetry group of
be denoted
ie: isotopy classes of diffeomorphisms of
which preserve
, where the isotopies are also required to preserve
.
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
![$ S^3 $](/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png)
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
![$ Sym(L) = \pi_0 Diff(S^3,L) $](/files/tex/4864a515a83902b1bc11af895975e9eb26387dda.png)
![$ S^3 $](/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png)
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
Now let be a hyperbolic link. Assume
has the further `Brunnian' property that there exists a component
of
such that
is the unlink. Let
be the subgroup of
consisting of diffeomorphisms of
which preserve
together with its orientation, and which preserve the orientation of
.
There is a representation given by restricting the diffeomorphism to the
. It's known that
is always a cyclic group. And
is a signed symmetric group -- the wreath product of a symmetric group with
.
Problem: What representations can be obtained?
Keywords: knot space; symmetry
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