Distinct Kissing number configurations How many essentially distinct kissing number configurations are there for the dimensions 3 & 4?  (I know the answer for dim 1,2,8,24). 
 A: Christine Bachoc and Frank Vallentin, in "New Upper Bounds For Kissing Numbers From Semidefinite Programming" (2007), indicate that

For dimension $3$, there are infinitely many possible configurations.  In the regular icosahedron configuration, the angular distances between the contact points are strictly greater than the required $\pi/3$; hence these points can be moved around obtaining infinitely many new suitable configurations.  This partially explains why the determination of $\tau_3$
  is difficult. On the contrary, uniqueness of the optimal configuration of points in dimension $4$ is widely believed, but it remains unproven.

This latter configuration refers to the lattice $D_4$, which yields the four-dimensional kissing number of $24$ (finally proven in 2003 by Oleg Musin, using a clever modification of the linear programming method devised by Philippe Delsarte and others).  I haven't found any reference that indicates that this uniqueness has been established in the last decade or so.
