Is there a well-known knot invariant that can be computed solely by inspecting an arbitrary projection of the knot into the plane (with a marking of each crossing as "over" or "under")?
The reason that the standard invariants don't suffice, and that these projections differ from standard knot diagrams, is that they allow the possiblity that one part of the knot is fully hidden behind another. For example, a projection that looks like a single unknotted loop can be obtained from the unknot, or from an unlink on arbitrarily many loops, manipulated in 3 dimensions so that their projections coincide in the plane. Thus any suitable invariant cannot distinguish between an unknot and an unlink, which e.g. rules out the Alexander-Conway polynomial (which is $1$ on the unknot but $0$ on any unlink).
Certain basic invariants like crossing number might work, but I'm more interested in a polynomial, homology, or similar.
The motivation for this question comes from an investigation of "fractional" Skein relations, in which crossings of a knot diagram can be resolved partially in one way and partially in another (whatever that means). When these fractions are rational, an alternate view is that each crossing is actually many crossings of hidden strands, which are then resolved differently.