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2003 Two applications of elementary knot theory to Lie algebras and Vassiliev invariants
Dror Bar-Natan, Thang T Q Le, Dylan P Thurston
Geom. Topol. 7(1): 1-31 (2003). DOI: 10.2140/gt.2003.7.1


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)U(g) for metrized Lie (super-)algebras g.


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Dror Bar-Natan. Thang T Q Le. Dylan P Thurston. "Two applications of elementary knot theory to Lie algebras and Vassiliev invariants." Geom. Topol. 7 (1) 1 - 31, 2003.


Received: 9 May 2002; Accepted: 8 November 2002; Published: 2003
First available in Project Euclid: 21 December 2017

zbMATH: 1032.57008
MathSciNet: MR1988280
Digital Object Identifier: 10.2140/gt.2003.7.1

Primary: 57M27
Secondary: 17B20 , 17B37

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

Rights: Copyright © 2003 Mathematical Sciences Publishers

Vol.7 • No. 1 • 2003
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