  In this note, we prove the existence of a tri-graded Khovanov-type bicomplex
(Theorem 1.2). The graded Euler characteristic of the total complex associated
with this bicomplex is the colored Jones polynomial of a link. The first
grading of the bicomplex is a homological one derived from cabling of the link
(i.e., replacing a strand of the link with several parallel strands); the
second grading is related to the homological grading of ordinary Khovanov
homology; finally, the third grading is preserved by the differentials, and
corresponds to the degree of the variable in the colored Jones polynomial. In
particular, we introduce a way to take a small cabling link diagram directly
from a big cabling link diagram (Theorem 3.2). This is a refined version of
arXiv:0907.5247.
