After giving the empirical considerations and the examples of non-invertible alternating knots with n 12 crossings, we developed the technique for obtaining non-invertible knots and determining the number of components of a KL from its Conway symbol. Our main goal was to find the general method for recognizing non-invertible knots directly from their Conway symbols. We succeed for some classes of knots, like alternating pretzel (stellar) knots, and extended this method to some other classes of algebraic and polyhedral knots.

 

As it was proved by L. Kauffman and S. Lambropoulou (2002a), all oriented rational KLs are invertible. Hence, the first non-invertible knots belong to stellar world.

For n=8:
          1) achiral non-invertible knot 817 .2.2 generates two-parameter family of non-invertible knots .(2p).(2q) (p,q 1), that are achiral non-invertible for p=q, and chiral non-invertible otherwise.
 

For n=9:

          2) non-invertible knot 932 .2 1.2 0 generates two-parameter family of non-invertible knots .(2p) 1.(2q) 0 (p,q 1);
          3) non-invertible knot 933 .2 1.2 generates two-parameter family of non-invertible knots .(2p) 1.(2q) (p,q 1).
 

For n=10:

          4) non-invertible knot 1067 2 2,3,2 1 generates four-parameter family of non-invertible knots (2p) (2q),(2r+1),(2s) 1 (p,q,r,s 1);
          5) achiral non-invertible knot 1079 (3,2) (3,2) generates four-parameter family of non-invertible knots ((2p+1),(2q)) ((2r+1),(2s)) (p,q,r,s 1), that are achiral non-invertible for p=r and q=s, and chiral non-invertible otherwise;
          6) non-invertible knot 1080 (3,2) (2 1,2) generates four-parameter family of non-invertible knots ((2p+1),(2q)) ((2r) 1,(2s)) (p,q,r,s 1);
          7) achiral non-invertible knot 1081 (2 1,2) (2 1,2) generates four-parameter family of non-invertible knots ((2p) 1,(2q)) ((2r) 1,(2s)) (p,q,r,s 1), that are achiral non-invertible for p=r and   q=s, and chiral non-invertible otherwise.
          Non-invertible knot 1082 .4.2 is the member of the family 1).
          8) Non-invertible knot 1083 .3 1.2 generates two-parameter family of non-invertible knots .(2p+1) 1.(2q) (p,q 1);
          9) non-invertible knot 1084 .2 2.2 generates three-parameter family of non-invertible knots .(2p) (2q).(2r) (p,q,r 1);
        10) non-invertible knot 1085 .4.2 0 generates two-parameter family of non-invertible knots .(2p).(2q) 0 for p q (p,q 1);
        11) non-invertible knot 1086 .3 1.2 0 generates two-parameter family of non-invertible knots .(2p+1).(2q) 0 (p,q 1);
        12) non-invertible knot 1087 .2 2.2 0 generates three-parameter family of non-invertible knots .(2p) (2q).(2r) 0 (p,q,r 1);
        13) achiral non-invertible knot 1088 .2 1.2 1 generates two-parameter family of non-invertible knots .(2p) 1.(2q) 1 (p,q 1), that are achiral non-invertible for p=q, and chiral non-invertible otherwise.
        14) non-invertible knot 1090 .3.2.2 generates three-parameter family of non-invertible knots .(2p+1).(2q).(2r) (p,q,r 1);

 

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