# Some Vassiliev Invariant Questions

1. 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.

2. 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.)

3. 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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retagged: general topology -> quantum topology. General topology means point-set topology: en.wikipedia.org/wiki/General_topology –  Daniel Moskovich May 31 '11 at 3:22

## 1 Answer

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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THX. 1) So Vassiliev invariants and undirected knots don't go together? 2) is then superfluous for my goal (but still, that link is a goldmine, many many THX!). Considering 3), the type number wasn't relevant. Could as well have used a higher order for the example: 0=S^3-3S+3S^-1-S^-3 or 0=S^4-4S^2+6S^0-4S^-2+S^-4 - I just wanted to know whether you can handle the definition as if it were an ordinary skein equation. –  Hauke Reddmann Jun 1 '11 at 14:09