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**
**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 L**_{1} and L_{2} are called *
P-undetectable* if P(L_{1})=P(L_{2}) 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+2t**^{2} and the Conway polynomial is
1-2 x.

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

**
**

Conjecture **
For every two alternating
non-isotopic KLs L**_{1} and L_{2} belonging to the same family
F, P(L_{1})
¹ P(L_{2}) for every polynomial invariant P.

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**

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
¹
q.

**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*.

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**

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, t_{1}=2p,
t_{2}=2q.

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