From the 637 non-invertible chiral knots, 68 of them are obtained
from the families already derived for n=8 and n=10. From the
remaining 569 knots, 554 are the generators of the new families of
chiral non-invertible knots without additional conditions for
parameters. In the following list are given the remaining 15 chiral non-invertible knots, the families of chiral non-invertible
knots derived from them, and the conditions for parameters:
| 12a76
| .4.2 1 0.2
| .(2p).(2q) 1 0.(2r)
| p
¹ r
|
| 12a192
| .4 1.2 1 0
| .(2p) 1.(2q) 1 0
| p
¹ q
|
| 12a201
| .4.2 1.2
| .(2p).(2q) 1.(2r)
| p
¹ r |
| 12a566
| 4 1,2 1 1,2 1
| (2p) 1,(2q) 1 1,(2r) 1
| p
¹ r
|
| 12a610
| 4:2 1:2
| (2p):(2q) 1:(2r)
| p
¹ r |
| 12a735
| 5,3,2 1+ |
(2p+1),(2q+1),(2r) 1+ |
p ¹ q |
| 12a753
| 5,2 1 1,3
| (2p+1),(2q) 1 1,(2r+1)
| p
¹ r |
| 12a782
| 2 1:4 0:2 0
| (2p) 1:(2q) 0:(2r) 0
| q
¹ r |
| 12a952
| 2.4.2 0.2
| (2p).(2q).(2r) 0.(2s)
| q
¹ s |
| 12a981
| .4.2.2 0.2 0
| .(2p).(2q).(2r) 0.(2s) 0
| p
¹ q |
| 12a984
| 4:3:2
| (2p):(2q+1):(2r)
| p
¹ r |
| 12a988
| 4:2:3 0
| (2p):(2q):(2r+1) 0
| p
¹ q |
| 12a1191
| 8*4:2
| 8*(2p):(2q)
| p
¹ q |
| 12a1238
| 3:4 0:2 0
| (2p+1):(2q) 0:(2r) 0
| q
¹ r
|
| 12a1240
| 4 0:3 0:2 0
| (2p) 0:(2q+1) 0:(2r) 0
| p
¹ r
|
Every alternating non-invertible knot (except those corresponding
to basic polyhedra) is the generator, or the member of the family
of non-invertible knots. Based on the properties of its generating
KL we can determine if some additional requirements are needed
for the whole family of KLs to be non-invertible. For example,
even though the generating knot 816 .2.2 0 is invertible,
the family it generates .(2p).(2q) 0 will contain only
non-invertible KLs beginning from the knot 1085 .4.2 0
provided that p ¹ q.
  
|