  We present a topological interpretation of knot and braid contact homology in
degree zero, in terms of cords and skein relations. This interpretation allows
us to extend the knot invariant to embedded graphs and higher-dimensional
knots. We calculate the knot invariant for two-bridge knots and relate it to
double branched covers for general knots.
  In the appendix we show that the cord ring is determined by the fundamental
group and peripheral structure of a knot and give applications.
