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<li>Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to <a href="http://front.math.ucdavis.edu/math.GT/0511602">my own paper</a> (joint with Tomotoda Ohtsuki), where we prove this for 3-loop Jacobi diagrams.</li>
<li>I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.</li>
<li>You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is <a href="http://uk.arxiv.org/PS_cache/q-alg/pdf/9702/9702009v1.pdf">this survey paper by Bar-Natan and Stoimenow</a>.</li>
<li>You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links ("Vassiliev Homotopy String Link Invariants" by Bar-Natan).</li>
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