23 research outputs found
Two applications of elementary knot theory to Lie algebras and Vassiliev invariants
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
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
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