The projection polynomial dD(t) can be directly transferred to link projections. In this case, the result is a polynomial of the form:
dD(t)=cntn+... +cktk,

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.

A similar polynomial, introduced somewhat earlier by C. Liang and Y. Jiang (1982), is effectively used by C. Liang and K. Mislow (1994a) for recognition of achiral knots. In that polynomial te(V) stands instead of e(V)t, and aij = s if the vertices i, j are connected with multiplicity s (s = 0,1,2). For achiral knot projections AD(t) = AD(t-1). For example, the achiral knot 2 2 (or 41) has

AD(t) = AD(t-1) = -4t2+8t-3+8t-1-4t-2.

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.

 

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