The projection polynomial d_{D}(t) can be directly transferred to
link projections. In this case, the result is a polynomial of the
form:
where n is the number of crossing points, and k is the number of link components. For every link, c_{n}=1. If a_{i} are link components, a_{ii}=w(a_{i}), and c_{k}=det(a_{ij}), where a_{ij}=lk(a_{i},a_{j}) denotes the linking number of the components a_{i}, a_{j}. 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 d_{D}(x,y,z)=x^{2}y^{2}z^{2}, 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 t^{e}^{(V}^{)} stands instead of e(V)t, and a_{ij} = s if the vertices i, j are connected with multiplicity s (s = 0,1,2). For achiral knot projections A_{D}(t) = A_{D}(t^{1}). For example, the achiral knot 2 2 (or 4_{1}) has
As they have pointed out, achirality is the result of antisymmetry (vertex signchanging symmetry), where a rotational antireflection produces invertible achiral knots, and a single 2antirotation produces noninvertible 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 Pdata.
