The V.I. definition goes doublepoint=overpass-underpass (or was it the other way around? If it's 50:50, I score 0 always :-). Would it lead anywhere to define doublepoint=overpass+underpass? (Even if it's equivalent. As I wouldn't have to guess then :-) I ask because I hate arrows on knots.
Stoimenow tabulated values for the 3-degree V.I., but only for knots. Has someone this data also for links? (Until 6 crossings already would be useful.)
Is this logic correct? Suppose you have a V.I. vanishing on all knots with >=2 doublepoints. Consider a clasp consisting of 2 doublepoints. From the definition then S^2+S^(-2)-2*S^0 = 0, where S is the usual braid generator resp. S matrix, and thus you can deduct a skein equation from the degree of a V.I. (and the higher the degree, the more of them).
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1) No. You can always mirror-reflect your definition across the plane that defines over/under. 2) I think so. Take a look in the Chmutov, Duzhin, Mostovoy survey on the arXiv. On page 92 they have the first ten non-trivial Vassiliev invariants, computed on sufficiently-many knots. If what you're looking for isn't near there, they likely have a reference for it. 3) Type one Vassiliev invariants are trivial for knots, and for links I believe linking number is the only non-trivial one. I'm not following your argument but I'm also not seeing where it's going. Perhaps this is what you're trying to prove? |
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