The projection invariant obtained is not the complete invariant of knot diagrams.

The polynomial d_D(t)=c_ntⁿ+... +c₁t has the following properties:

1. for every alternating knot projection D, the degree of d_D(t) is n, |c_n|=1 and |c₁|=|w(D)|, where w(D) is the writhe of D;
2. d_D(t) and d_D(-t) correspond to obverse (mirror-symmetrical) knot diagrams;
3. for n=0 (mod 2), a change of the orientation results in a change of d_D(t) to d_D(-t), and for n=1 (mod 2) in a change of d_D(t) to -d_D(-t).

According to (2) and (3), in the set of all polynomials d_D(t) we may distinguish even polynomials (d_D(t)=d_D(-t)), containing only even degrees of t, corresponding to achiral knot projections, and odd polynomials (d_D(t)=-d_D(-t)), containing only odd degrees of t, which are invariant to a change of orientation. As with every polynomial invariant, the projection polynomial d_D(t) sometimes fails to detect isomorphism of knot projections or achirality. For example, for n=10 from 364 non-isomorphic projections of alternating knots it recognizes 363 of them as different, and sometimes fails to detect achirality (that is, sometimes yields an even polynomial d(t) for a chiral knot projection).

The following table contains polynomials d_D(t) for all different projections of knots with n £ 7 crossings. In the first column the symbol of a knot is given in the classic notation, followed by the Conway symbol of the corresponding projection, minimal Dowker code and the projection polynomial d_D(t) given by the sequence of its coefficients in descending order.