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

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

Geometry & Topology 7 (2003) 1–31

arXiv: math.QA/0204311


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.

Wheels, Wheeling, Vassiliev invariants, Hopf link, $1+1=2$, Duflo isomorphism, cabling
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 17B20, 17B37
Forward citations
Received: 9 May 2002
Accepted: 8 November 2002
Published: 23 January 2003
Proposed: Vaughan Jones
Seconded: Yasha Eliashberg, Joan Birman
Dror Bar-Natan
Department of Mathematics
University of Toronto
M5S 3G3
Thang T Q Le
Department of Mathematics
SUNY at Buffalo
New York 14214
Dylan P Thurston
Department of Mathematics
Harvard University
Massachusetts 02138