470 research outputs found
Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary
Let be the -punctured disk. We prove that a family of essential
simple arcs starting and ending at the boundary and pairwise intersecting at
most twice is of size at most . On the way, we also show that
any nontrivial square complex homeomorphic to a disk whose hyperplanes are
simple arcs intersecting at most twice must have a corner or a spur.Comment: Assaf Bar-Natan's MSc thesis, written under the supervision of Prof.
Piotr Przytyck
A Note on the Unitarity Property of the Gassner Invariant
We give a 3-page description of the Gassner invariant / representation of
braids / pure braids, along with a description and a proof of its unitarity
property.Comment: Mistake fixed in the accompanying Mathematica notebook (not the paper
itself
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
- …