# nLab Adams operation on Jacobi diagrams

Contents

### Context

#### Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

# Contents

## 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.

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:

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

## References

Last revised on January 3, 2020 at 06:21:51. See the history of this page for a list of all contributions to it.