Experimenting with KLs and computing their polynomials, you can verify some of their well known and already proved properties. For example, to every alternating KL corresponds an Alexander polynomial with alternating signs. Hence, the Alexander polynomial of any alternating KL cannot be equal to 1, but there are many non-trivial KLs with the Alexander and Conway polynomials equal to 1, belonging to infinite families of KLs with the same property. Many different KLs will have equal Alexander polynomials. Alexander polynomials do not distinguish a KL from its mirror image. For Alexander polynomials the relation A(K1#K2) = A(K1)A(K2) holds. This property holds for Jones polynomials (and HOMFLYPT polynomials) as well: V(K1#K2) = V(K1)V(K2). For Jones polynomials of a knot or link L and its mirror image [`L] the relation V[`L](t) = VL(t-1) holds. The HOMFLYPT polynomial of [`L] is obtained by replacing each l in the HOMFLYPT polynomial of L by l-1. A Jones polynomial (or HOMFLYPT) polynomial of a KL remains the same after changing orientations of its components. The bracket polynomial of an achiral KL must be palindromic.

Despite all nice properties and the increasing sensitivity of the polynomial invariants, sometimes they may lead to wrong conclusions. For example, the non-trivial link

9*5 1 2:-1.-1.2.-1.-1:-5 -1 -2
has a trivial Jones polynomial. Moreover, there is a whole class of such links (Eliahou, Kauffman and Thistlethwaite, 2003).

In detecting chirality polynomials sometimes fail, even all at the same time: for instance, Jones, HOMFLYPT, and Kauffman polynomials calculated for the knots 2 2,3,-2 (942), (2 1,2+) (3,-2), or 10**2 0.2.2.2 0.2 0.2 0 and their mirror images are equal. However, those knots are chiral. Moreover, this property holds for the whole families of alternating chiral KLs 10***p::.p 0, 10***p::.p 0:.q 0.q, 10***p::.p 0:.q 0.q, 10***p::.p 0:.q.q 0, 10***p:.q 0.q.p 0, 10***p.q 0:.q:.p 0.r 0.r, 10***p:.q 0.q.p 0:.r.r 0 etc.

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