Riordan trees and the homotopy sl2 weight system

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 Uq(sl2)Uq(sl2), 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 sl2sl2 weight system, which is its restriction to the space of Jacobi diagrams labeled by distinct integers. Noting that the sl2sl2 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.