Recall that a link is an embedding of some number of circles in space up to isotopy. (Think "a knot but maybe multiple pieces.") Three oriented links form a skein triple if they have identical diagrams except for at one place, where they differ as in this image:
(This image should be familiar to anyone who has worked with skein relations, such as those used to define the HOMFLY polynomial.)
Suppose $F$ is a function from the set of oriented links to $\Bbb R^n$. We make the following definition:
Definition: A function $F:\{\text{oriented links}\}\to\Bbb R^n$ has the skein collinearity condition (SCC) if $F(L_-)$, $F(L_0)$, and $F(L_+)$ are collinear for any skein triple $L_-$, $L_0$, and $L_+$.
Note that the domain is oriented links, not oriented link diagrams, meaning that $F$ is required to be a link invariant. For example, this image shows that for any $F$ satisfying the SCC, $F(\text{trefoil})$, $F(\text{Hopf link})$, and $F(\text{unknot})$ are collinear.
Any function $F$ whose image lies on a single line satisfies the SCC. Are there any nontrivial solutions? That is, are there any functions $F$ satisfying the SCC such that there are three links whose images are not collinear?
(PS The proper way to phrase this question probably involves matroids, which were invented to axiomatize "independence" phenomena such as collinearity and coplanarity, but I'm not sure the right way to phrase it.)