  We define K-stability of a polarized Sasakian manifold relative to a maximal
torus of automorphisms. The existence of a Sasaki-extremal metric in the
polarization is shown to imply that the polarization is K-semistable. Computing
this invariant for the deformation to the normal cone gives an extention of the
Lichnerowicz obstruction, due to Gauntlett, Martelli, Sparks, and Yau, to an
obstruction of Sasaki-extremal metrics. We use this to give a list of examples
of Sasakian manifolds whose Sasaki cone contains no extremal representatives.
These give the first examples of Sasaki cones of dimension greater than one
that contain no extremal Sasaki metrics whatsoever. In the process we compute
the unreduced Sasaki cone for an arbitrary smooth link of a weighted
homogeneous polynomial.
