Two applications of elementary knot theory to Lie algebras and Vassiliev invariants

Abstract

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures, which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of $1+1=2$ on an abacus. The Wheels conjecture is proved from the fact that the $k$–fold connected cover of the unknot is the unknot for all $k$.

Along the way, we find a formula for the invariant of the general $\left(k,l\right)$ cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo–Kirillov map $S\left(\mathfrak{g}\right)\to U\left(\mathfrak{g}\right)$ for metrized Lie (super-)algebras $\mathfrak{g}$.