The projection polynomial dD(t) can be directly transferred to
link projections. In this case, the result is a polynomial of the
where n is the number of crossing points, and k is the number of link components. For every link, |cn|=1. If ai are link components, aii=w(ai), and |ck|=|det(aij)|, where aij=lk(ai,aj) denotes the linking number of the components ai, aj. In order to increase the selectivity of this polynomial, it is possible to denote generators belonging to different components of a link by different variables. For example, Borromean rings have projection polynomial dD(x,y,z)=x2y2z2, which implies that their components are interchangeable, and that Borromean rings are (probably) achiral.
As they have pointed out, achirality is the result of antisymmetry (vertex sign-changing symmetry), where a rotational antireflection produces invertible achiral knots, and a single 2-antirotation produces non-invertible achiral knots. Achiral knots are members of infinite families.
The LinKnot functions JablanPoly (webMathematica JablanPoly) and LiangPoly (webMathematica LiangPoly) calculate these two polynomial invariants for a KL projection given by its Conway symbol, Dowker code, or P-data.