Vassiliev-Kontsevich invariants and Parseval's theorem

TL;DR

The paper explores the relationship between Vassiliev-Kontsevich invariants and knot reconstruction, using Fourier analysis and Parseval's theorem to support the idea that these invariants can uniquely determine knots.

Contribution

It provides evidence that Vassiliev-Kontsevich invariants can be redefined to reconstruct knots as a power series, introducing a novel approach involving Fourier analysis and braid group problems.

Findings

Proof of $e^ au=q$ using Parseval's theorem

Representation of $ au$ as a Laurent series

Discussion of a Plancherel theorem for braid groups

Abstract

We use an example to provide evidence for the statement: the Vassiliev-Kontsevich invariants $k_n$ of a knot (or braid) $k$ can be redefined so that $k = \sum_0^\infty k_n$. This constructs a knot from its Vassiliev-Kontsevich invariants, like a power series expansion. The example is pure braids on two strands $P_2\cong \mathbb{Z}$, which leads to solving $e^\tau=q$ for $\tau$ a Laurent series in $q$. We set $\tau = \sum_1^\infty (-1)^{n+1} (q^n - q^{-n})/n$ and use Parseval's theorem for Fourier series to prove $e^\tau=q$. Finally we describe some problems, particularly a Plancherel theorem for braid groups, whose solution would take us towards a proof of $k=\sum_0^\infty k_n$.