The Kontsevich Integral in Book Notation

TL;DR

This paper introduces a novel matrix-based book notation for the Kontsevich integral of tangles, enabling algebraic manipulation of moves and invariants, and provides a method to recover the original link from its invariant.

Contribution

It develops a matrix representation of tangle chord diagrams called books, facilitating algebraic operations and link reconstruction from the Kontsevich integral.

Findings

Matrix representation of chord diagrams as books.

Implementation of moves via matrix congruences.

Ability to recover the original link from the invariant.

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.