Given the diagram of an unoriented knot $K \subset S^3$, one can compute its Jones polynomial by picking an orientation for the knot and then simplifying the diagram using the relations drawn in the "properties" section here (the choice of orientation doesn't matter, although presumably it matters for links). Alternatively, one can give the knot a 0 framing and then simplify using the relations drawn here.
The HOMFLY polynomial of $K$ is computed in a way similar to the first method of computing the Jones polynomial - pick an orientation of the knot and simplify the diagram using the relations here.
Is there a way of computing the HOMFLY polynomial of $K$ analogous to the second way of computing the Jones polynomial described above, i.e. give $K$ the 0 framing (but no orientation) and simplify the diagram using some relations?