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 $e^{\tau }=q$ 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 $k={\sum }_0^{\infty }{k}_n$.

5 pages, 2 figures. Extensively revised. Discussion of extending result to braids on more strands and to knots added. Two figures added