The Kontsevich Integral in Book Notation
Abstract
We introduce a matrix representation of a chord on a tangle which leads us to representing tangle chord diagrams as stacks of matrices that we call books. We show that band sum moves, Reidemeister moves as well as orientation changes are implemented on \widetilde{Z}_f - a framed link invariant constructed from the Kontsevich integral that's well-behaved under band sum moves - by matrix congruences. We prove that being given the bare framed Kontsevich integral Z_f(L) in book notation for some unknown link L, we can determine what the link L is, as well as the projection of Z_f(L) in the original completed algebra of chord diagrams.
Cite
@article{arxiv.1010.2814,
title = {The Kontsevich Integral in Book Notation},
author = {Renaud Gauthier},
journal= {arXiv preprint arXiv:1010.2814},
year = {2010}
}
Comments
81 pages, "punct" has been replaced by "M-resolved", invariance of Z_f on the choice of normalization added in section 2.4, as well as a more precise definition of the doubling map. Proposition 7.16 added at the very end. Typos corrected. Reference added