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,], 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



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