2.9  Families of undetectable KLs

The conversion to PD (planar diagrams) that are the main input for the program KnotTheory provide LinKnot functions fConwayToPD (webMathematica fConwayToPD), fKnotscapeDowToPD (webMathematica fKnotscapeDowToPD), fDowkerToPD (webMathematica fDowkerToPD), and fPdataToPD (webMathematica fPdataToPD). The minimum braid words can be obtained by the function BR (webMathematica BR), fBraidW (webMathematica fBraidW) gives the corresponding braid word, and fPDfromBW (webMathematica fPDfromBW) converts P-data to braid word.

We will consider the set of polynomial invariants P, where P is Alexander, Conway, Jones, Khovanov, A2, Links-Gould- HOMFLYPT, Kauffman, or colored Jones polynomial. In this research, Khovanov, A2, and colored Jones polynomials are computed by using additional LinKnot functions Kh (webMathematica Kh), A2Invariant (webMathematica A2Invariant), and ColouredJones (webMathematica ColouredJones) from the program KnotTheory, and Links-Gould invariants are computed by using the additional functions LinksGould (webMathematica LinksGould) and LinksGouldInv (webMathematica LinksGouldInv) from the program Links-Gould Explorer written by David de Wit. To the original function for computing Links-Gould invariants are added options to use as input p-data, Conway symbol, or Dowker code of KL.

One of the main questions about every polynomial invariant is whether or not exists a P-unknot (unlink), that is, a nontrivial knot (link) L with the trivial polynomial P(L). For example, Alexander and Conway unknots and Jones unlinks exist (Eliahou, Kauffman and Thistlethwaite, 2003). They are not single KLs, but the families of KLs with this property. The question about Jones unknot still remains open, but till now it is proved that, if it exists, Jones unknot must be non-alternating knot (Murasugi, 1987a,b) with at least n=18 crossings (Dasbach and Hougardy, 1997). For the remaining P-invariants the question about P-unknot is open as well.

Two non-isotopic knots or links L1 and L2 are called P-undetectable if P(L1)=P(L2) for some polynomial invariant P. There are many P undetectable KLs with the same or different number of crossings, and there is an infinite number of undetectable KLs for any polynomial invariant P.

Moreover, there are complete families of KLs that cannot be detected by certain polynomial invariant P: for every knot or link L from such a family F the result is the same, i.e., P(L) is the same for all KLs from F. We already mentioned that, e.g., for every knot from the family (2k+1),3,-3 the Alexander polynomial is 2-5t+2t2 and the Conway polynomial is 1-2 x.

For families of alternating KLs, we propose the following Conjecture:

For every two alternating non-isotopic KLs L1 and L2 belonging to the same family F, P(L1) P(L2) for every polynomial invariant P.

In order to avoid overlapping, we must be very careful when defining families. For example, 2p 2p will be considered as one-parameter family, and 2p 2q as two-parameter family with p

With non-alternating KLs the situation is absolutely different. The family F of non-alternating KLs will be called P-undetectable if P(L) is the same for any link L from F. If a polynomial P can recognize as different any two non-isotopic KLs from F, we will call this family P-detectable.

Conjecture Every Alexander-undetectable family is Conway-undetectable and vice versa.

An interesting class of examples was due to K. Kanenobu (1986), who gave infinitely many examples of knots that are Jones- and/or HOMFLYPT-undetectable. Kanenobu's examples can be reconstructed in the framework of S. Eliahou, L. Kauffman, and M. Thistlethwaite (2003), and this was done, e.g., by L. Watson (2005). K. Luse and Y. Rong (2006) constructed a new family of Jones- and HOMFLYPT-undetectable knots. Kanenobu's knots can be obtained from this family for a=1, t1=2p, t2=2q.