23 research outputs found

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

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    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.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper1.abs.htm

    Abstracts from the 20th International Symposium on Signal Transduction at the Blood-Brain Barriers

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    https://deepblue.lib.umich.edu/bitstream/2027.42/138963/1/12987_2017_Article_71.pd

    PROOF OF A CONJECTURE OF KULAKOVA ET AL. RELATED TO THE sl2 WEIGHT SYSTEM

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    Abstract. In this article, we show that a conjecture raised in [KLMR], which regards the coefficient of the highest term when we evaluate the sl2 weight system on the projection of a diagram to primitive elements, is equivalent to the Melvin-Morton-Rozansky conjecture, proven in [BG]. 1
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