#### Geometry & Topology, Vol. 7 (2003)
Paper no. 1, pages 1--31.

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

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

**Abstract**.
Using elementary equalities between various cables of the unknot and
the Hopf link, we prove the Wheels and Wheeling conjectures of
[Bar-Natan, Garoufalidis, Rozansky and Thurston, arXiv:q-alg/9703025]
and [Deligne, letter to Bar-Natan, January 1996,
http://www.ma.huji.ac.il/~drorbn/Deligne/], 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)-->U(g) for metrized Lie (super-)algebras g.
**Keywords**.
Wheels, Wheeling, Vassiliev invariants, Hopf link, $1+1=2$, Duflo isomorphism, cabling

**AMS subject classification**.
Primary: 57M27.
Secondary: 17B20, 17B37.

**DOI:** 10.2140/gt.2003.7.1

**E-print:** `arXiv:math.QA/0204311`

Submitted to GT on 9 May 2002.
Paper accepted 8 November 2002.
Paper published 23 January 2003.

Notes on file formats
Dror Bar-Natan

Department of Mathematics, University of Toronto

Toronto ON M5S 3G3, Canada

Thang TQ Le

Department of Mathematics, SUNY at Buffalo

Buffalo NY 14214, USA

Dylan P Thurston

Department of Mathematics, Harvard University

Cambridge, MA 02138, USA

Email: drorbn@math.toronto.edu, letu@math.buffalo.edu, dpt@math.harvard.edu

URL:
http://www.math.toronto.edu/~drorbn,
http://www.math.buffalo.edu/~letu,
http://www.math.harvard.edu/~dpt

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