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
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.
Wheels, Wheeling, Vassiliev invariants, Hopf link, $1+1=2$, Duflo isomorphism, cabling
AMS subject classification.
Secondary: 17B20, 17B37.
Submitted to GT on 9 May 2002.
Paper accepted 8 November 2002.
Paper published 23 January 2003.
Notes on file formats
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: email@example.com, firstname.lastname@example.org, email@example.com
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