nLab Adams operation on Jacobi diagrams

Idea

On the vector space $\mathcal{A}$ of Jacobi diagrams modulo STU-relations (equivalently chord diagrams modulo 4T relations) there is a system of linear maps (for $q \in \mathbb{Z}$, $q \neq 0$)

$$\psi^q \;\colon\; \mathcal{A} \longrightarrow \mathcal{A}$$

which respect the coalgebra structure and satisfy

$$\psi^{q_2} \circ \psi^{q_1} \;=\; \psi^{q_1 \cdot q_2}$$

and as such are (dually) analogous to the Adams operations on topological K-theory.

(Bar-Natan 95, Def. 3.11 & Theorem 7)

In fact, when evaluated in Lie algebra weight systems $w_{\mathbf{N}}$ and under the identification (see here) of the representation ring of a compact Lie group $G$ with the $G$-equivariant K-theory of the point, these Adams operations on Jacobi diagrams correspond to the Adams operations on equivariant K-theory:

$$w_{\mathbf{N}}(\psi^q D) \;=\; w_{\psi^q \mathbf{N}}(D) \,.$$

(Bar-Natan 95, Exc. 6.24 )

References

• Dror Bar-Natan, On the Vassiliev knot invariants, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, pdf)