Vassiliev-Kontsevich invariants and Parseval's theorem

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 $e^τ=q$ for $τ$ a Laurent series in $q$. We set $τ= \sum_1^\infty (-1)^{n+1} (q^n - q^{-n})/n$ and use Parseval's theorem for Fourier series to prove $e^τ=q$. Finally we describe some problems, particularly a Plancherel theorem for braid groups, whose solution would take us towards a proof of $k=\sum_0^\infty k_n$.