Volume 12, issue 1 (2012)

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Noninjectivity of the “hair” map

Bertrand Patureau-Mirand

Algebraic & Geometric Topology 12 (2012) 415–420
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.

Keywords
finite type invariant, Feynman diagram
Mathematical Subject Classification 2010
Primary: 57M25, 57M27