Riordan trees and the homotopy $sl_2$ weight system

TL;DR

This paper introduces a modified dual canonical basis for invariant tensors related to quantum topology and uses it to analyze the homotopy $sl_2$ weight system, computing its action on trees and discussing its kernel.

Contribution

It presents a new modified basis for invariant tensors and applies it to study the homotopy $sl_2$ weight system, providing explicit computations and kernel analysis.

Findings

Computed the image of the homotopy $sl_2$ weight system on connected trees in all degrees.

Discussed the kernel of the homotopy $sl_2$ weight system.

Introduced a modified dual canonical basis in terms of Jacobi diagrams.

Abstract

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.