Noninjectivity of the “hair” map

Abstract

Kricker constructed a knot invariant $Z^{rat}$ valued in a space of Feynman diagrams with beads. When composed with the “hair” map $H$, it gives the Kontsevich integral of the knot. We introduce a new grading on diagrams with beads and use it to show that a nontrivial element constructed from Vogel’s zero divisor in the algebra $\Lambda$ is in the kernel of $H$. This shows that $H$ is not injective.