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We extended their
construction to the four-parameter family F
of the form 6*-a.-b.t1 x 0.b.a.t2 y 0,
where are a, b are fixed positive integers, t1 and t2
are arbitrary fixed tangles, x, y are integers, and const is a constant. This family has the following properties:
- all KLs from F are Alexander- and
Conway-undetectable for every x, y satisfying the condition x=y (mod 2);
- all KLs from F are Jones- and
Khovanov-undetectable for every x, y satisfying the condition
|x-y|=const, where const is an even constant;
- all KLs from F are HOMFLYPT-, A2-,
and Links-Guld-undetectable for every pair of even numbers x, y
satisfying the condition |x-y|=const;
- all KLs from F are HOMFLYPT-, A2-,
and Links-Guld-undetectable for every pair of odd numbers x, y
satisfying the condition |x-y|=const.
All undetectable KLs
obtained are Kauffman- and colored Jones-detectable. KLs obtained are
achiral for t1=t2, and chiral otherwise.
The results are summarized
in the following table:
| Invariant
|
(1)
|
(2)
|
(3) or (4)
|
| Alexander
|
- |
- |
- |
| Conway
|
- |
- |
- |
| Jones
|
+ |
- |
- |
| Khovanov
|
+ |
- |
- |
| HOMFLYPT
|
+ |
+ |
- |
| A2
|
+ |
+ |
- |
| Links-Gould
|
+ |
+ |
- |
| Kauffman
|
+ |
+ |
+ |
| Colored Jones
|
+ |
+ |
+ |
For example, for
a=b=2, (x,y) Î
{(2,6),(2,2),(3,3),(4,4),(5,5)} are obtained the following knots:
| K1=6*-2.-2.2 0.2.2.6 0
|
K2=6*-2.-2.2 0.2.2.2 0
|
| K3=6*-2.-2.3 0.2.2.3 0
|
K4=6*-2.-2.4 0.2.2.4 0
|
| K5=6*-2.-2.5 0.2.2.5 0
|
|
All of them are Alexander and
Conway-undetectable; K1 can be recognized as different from the
other four by all the other polynomials; K2 and K3, or
K4 and K5 can be distinguished by HOMFLYPT, A2 or
Links-Gould invariant, but they don't distinguish K2 from K4,
or K3 from K5. All of them can be distinguished only
by Kauffman, or colored Jones polynomial.
  
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