I don't have much knowledge in either knots or Hopf bialgebras, but I think one can answer part of your question (the one formerly asked as a title: "What do negative knots look like?") anyway.
About finite type invariants: Wikipedia says it is still not known whether finite type invariants separate knots (and I think a discovery of an answer to this question would make pretty big news anyway).
It matters whether they do or not if you want to learn about all points of $V$. However, either way you can be certain that the points of $V$ (which I'll denote by $\mathop{\text{Spec}} V$) can be formally added or subtracted. In other words, there are points in $\mathop{\text{Spec}} V$ corresponding to $[\text{unknot}]+2\cdot[\text{trefoil}]$ or $-[\text{trefoil}]$. This is implied when you say that those form Hopf bialgebra.
So, for any given knot $K$ there exists an element of $\mathop{\text{Spec}} V$ that has finite invariants negative of finite invariants of knot $K$, but this observation is rather trivial.
Thus I think one shouldn't ask what is asked in the title, but rather more general question (emphasized); update: Theo Johnson-Freyd did so, rendering the answer rather irrelevant.