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

By Dror Bar-Natan, Thang T. Q. Lê, Dylan Thurston. 2002

90 citations recorded by the source corpus.

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [5, 9], 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 nd 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. AMS Classication numbers Primary: 57M27 Secondary: 17B20, 17B37

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2f35c3502c91eb74e0520d79590a32ee3d9b7af7

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Citations in the corpus

Lectures on Duflo isomorphisms in Lie algebra and complex geometry (2011, background): “[4] D. Bar-Natan, T. Le, and D. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom.”
Problems on invariants of knots and 3-manifolds (2004, methodology): “The Kontsevich invariant of a cable knot of a knot K can be calculated by applying a cabling formula [38] to the Kontsevich invariant of K .”
On the Rozansky-Witten weight systems (2006, methodology): “3 (Wheeling theorem [5]) The composition of the wheeling map \ with the symmetrisation map is an algebra isomorphism B!A.”
Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial (2014, background): “Thus, the first named author, DBN, was absolutely delighted when in January 2008 Anton Alekseev described to him his joint work [2] with Charles Torossian: Anton told DBN that they found a relationship between the Kashiwara–Vergne conjecture [52], a cousin of the Duflo isomorphism (which DBN already”
Generalized trace and modified dimension functions on ribbon categories (2010, background): “The path is especially well trodden going from algebra to topology (although notable exceptions do exist [6]).”