On geometric link bases for A-polynomials

A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum A-polynomials for knots. We treat A-polynomials as relations between different links, obtained by hanging additional "simple" components on the original knot. Depending on the choice of this "decoration", the knot polynomial is either multiplied by a number or decomposes into a sum over "surrounding" representations by a cabling procedure. What happens is that these two of decorations, when complicated enough, become dependent – and this provides an equation. To make these geometric methods somewhat simpler we suggest to use an arcade formalism/representation of the braid group to simplify decorating links universally. However, in this framework the eventual equivalence of links is not a topological property – it follows from relations among R-matrices, and depends on the choice the gauge group and incorporates specific link graph relations known as brackets: in practice we will consider only the Kauffman bracket for SU(2) and the Kuberberg bracket for SU(3), however a generalization to SU(n) is potentially available. In a quasi-classical limit it is closely related to the well publicized augmentation theory and contact geometry. The talk is based on papers 2408.08181 and 2505.20260 with A.Morozov.