Vassiliev-Kontsevich invariants and Parseval's theorem

Details Vassiliev-Kontsevich invariants and Parseval's theorem Vassiliev-Kontsevich invariants and Parseval's theorem

2009-09-28

arxiv.org/abs/0909.5178v2

Mathematics

Quantum Algebra

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

Scientific paper

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 $k=\sum_0^\infty k_n$.

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