  In this paper, we investigate abelian knot contact homology by using the
trace-free slice of a knot, which is the cross-section of the character variety
of a knot cut by the hyperplanes defined by meridionally trace-free (traceless)
conditions. We first give a set of equations whose common solutions coincide
with the trace-free slice. This gives us a strong tool to describe exactly the
relationship between the trace-free slice, degree 0 abelian knot contact
homology and the character variety of the 2-fold branched cover of the 3-sphere
branched along the knot. Using this relationship, we introduce a notion called
ghost characters of a knot to check the followings: (1) when the conjecture
given by L. Ng concerned with the above relationship does not hold, (2) when
the map from the trace-free slice to the character variety of the 2-fold
branched cover, constructed by Y. Yamaguchi and the author, is not surjective.
Note that in the recent paper of Christopher R. Cornwell, he has translated the
results on the above relationship into the language of "reflective
augmentation".
