Achiral knots with n=12 crossings were derived by M.G. Haseman (1918). In her original work, all of them except one (H59=H60) are shown by their centro-antisymmetrical presentations. After correction for the knot 10. 2 2 2 2 2 2 (shortly written as 26), and identification of 7 duplications (Thistlethwaite, 1985), the complete list contains 54 knots. They are given by Conway symbols showing their (anti)symmetry:

01.  62     16.   = 13.       31.  12K     46.  8*2 0.2 0.2 0.2 0
02.  5 12 5     17.  (2 2,2)2     32.  8*2.2.2 0::.2 0     47.  3 0.2.2.3 0
03.  (3,3)2     18.  2 42 2     33.  8*.2 1.2 1     48.  8*3:.3
04.  3 2 12 2 3     19.  2.2.2.2.2 0.2 0     34.  8*2.2:.2 0.2 0     49.  8*3 0:.3 0
05.  34     20.  2.2 1.2 0:2 1     35.  10**2 0::.2 0     50.  .2 2.2 2
06.  (2 1,2 1)2     21.  2.2.2 0.2.2.2 0     36.   = 20.       51.   = 50.  
07.  3 1 22 1 3     22.  12L     37.  8*2 1 0:.2 1 0     52.  .3.3.2 0.2 0
08.  22 14 22     23.  10**:2 0::.2 0     38.  .4.4     53  10*2.2
09.  2 18 2     24.  10*2 0::.2 0     39.  .3 1.3 1     54.   = 40.
10.  26     25.  8*2:2 0.2 0:2    40.  8*2.2:.2.2     55.  10**:2 0.2 0
11.  4 22 4     26.  8*2.2.2.2     41.  8*3.3     56.  10**:2.2
12.  2 1 2 12 2 1 2     27.  2.2 1.2 1.2     42.  8*3 0.3 0     57.   = 45.
13.  (2 1 1,2)2     28.  .2 1.2 1.2.2     43.  10*2:::.2     58.  12B
14.  (3,2+)2     29.  3.2.2.3     44.  10**:2::.2     59.  .2 1.2.2 1 0.2 0
15.  (2 1,2+)2     30.  .3.3.2.2     45.  10**2::.2     60.   = 59.
            61.   = 6.  

In the case of the P-world for n £ 12, we will restrict the discussion of achirality to the basic polyhedra and alternating KLs generated from them as families. We conjecture that alternating achiral KLs can only be derived from achiral basic polyhedra or achiral source KLs by an arrangement of tangles preserving achirality.

We already mentioned that some more sensitive polynomial invariants (e.g., Jones, Kauffman or HOMFLYPT polynomials) detect chirality, but not always: sometimes they will identify a certain KL that is not achiral as achiral.

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