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

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 $(k, l)$ cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo–Kirillov map $S (g) \to U (g)$ for metrized Lie (super-)algebras $g$ .

Keywords

Wheels, Wheeling, Vassiliev invariants, Hopf link, $1+1=2$, Duflo isomorphism, cabling

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 17B20, 17B37

References

Forward citations

Publication

Received: 9 May 2002

Accepted: 8 November 2002

Published: 23 January 2003

Proposed: Vaughan Jones

Seconded: Yasha Eliashberg, Joan Birman

Authors

Dror Bar-Natan
Department of Mathematics University of Toronto Toronto Ontario M5S 3G3 Canada
http://wwww.math.toronto.edu/~drorbn/
Thang T Q Le
Department of Mathematics SUNY at Buffalo Buffalo New York 14214 USA
http://www.math.buffalo.edu/~letu/
Dylan P Thurston
Department of Mathematics Harvard University Cambridge Massachusetts 02138 USA
http://www.math.harvard.edu/~dpt/

Volume 7, issue 1 (2003)

Dror Bar-Natan, Thang T Q Le and Dylan P Thurston