What is it about?
As the title says, we prove that a certain space of diagrams vanishes. This shows that there are no Vassiliev invariants (finite type invariants) which distinguish a knot from its reverse in the corresponding class of invariants.
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Why is it important?
Can finite type invariants distinguish prime knots? The question essentially boils down to whether they can distinguish a knot from the same knot with reversed orientations. It's clear for reasons of symmetry that 0-loop, 1-loop, and 2-loop finite type invariants cannot. We prove here that 3-loop finite type invariants cannot either.
Perspectives
The 4-loop case is open, tractable, and interesting, and somebody should work it out. I am personally convinced that, for sufficiently high loop degree (5 or 6 should be enough), this space of Jacobi diagrams will not be empty, and somebody will find a Vassiliev invariant which distinguishes a knot from its reverse . Note also that Victor Tourchine and collaborators have extended our method to the wider context of embedding spaces using Goodwillie Calculus, and have some very nice and deep results in this direction.
Dr Daniel Moskovich
Ben-Gurion University of the Negev
Read the Original
This page is a summary of: Vanishing of 3-loop Jacobi diagrams of odd degree, Journal of Combinatorial Theory Series A, July 2007, Elsevier,
DOI: 10.1016/j.jcta.2006.10.005.
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