  The purpose of this paper is twofold. On one hand, we introduce a
modification of the dual canonical basis for invariant tensors of the
3-dimensional irreducible representation of $U_q(sl_2)$, given in terms of
Jacobi diagrams, a central tool in quantum topology. On the other hand, we use
this modified basis to study the so-called homotopy $sl_2$ weight system, which
is its restriction to the space of Jacobi diagrams labeled by distinct
integers. Noting that the $sl_2$ weight system is completely determined by its
values on trees, we compute the image of the homotopy part on connected trees
in all degrees; the kernel of this map is also discussed.
