  In a single state of a quantum dot the Kondo effect arises due to the
spin-degeneracy, which is present if the dot is occupied with one electron (N =
1). The eigenstates of a carbon nanotube quantum dot possess an additional
orbital degeneracy leading to a four-fold shell pattern. This additional
degeneracy increases the possibility for the Kondo effect to appear. We revisit
the Kondo problem in metallic carbon nanotubes by linear and non-linear
transport measurement in this regime, in which the four-fold pattern is
present. We have analyzed the ground state of CNTs, which were grown by
chemical vapor deposition, at filling N = 1, N = 2, and N = 3. Of particular
interest is the half-filled shell, i.e. N = 2. In this case, the ground state
is either a paired electron state or a state for which the singlet and triplet
states are effectively degenerate, allowing in the latter case for the
appearance of the Kondo effect. We deduce numbers for the effective missmatch d
of the levels from perfect degeneracy and the exchange energy J. While d ~ 0.1
- 0.2 (in units of level spacing) is in agreement with previous work, the
exchange term is found to be surprisingly small: J < 0.02. In addition we
report on the observation of gaps, which in one case is seen at N = 3 and in
another is present over an extended sequence of levels.
