d(t)=



t
1
-1
0
0
-t
1
-1
-1
0
t
1
1
-1
0
-t




=t4-2t2.

 

If dD'(t) dD"(t), two oriented knot projections D' and D" are non-isomorphic (Jablan, 1995).

The projection invariant obtained is not the complete invariant of knot diagrams.

The polynomial dD(t)=cntn+... +c1t has the following properties:

  1. for every alternating knot projection D, the degree of dD(t) is n, |cn|=1 and |c1|=|w(D)|, where w(D) is the writhe of D;
  2. dD(t) and dD(-t) correspond to obverse (mirror-symmetrical) knot diagrams;
  3. for n=0 (mod 2), a change of the orientation results in a change of dD(t) to dD(-t), and for n=1 (mod 2) in a change of dD(t) to -dD(-t).

According to (2) and (3), in the set of all polynomials dD(t) we may distinguish even polynomials (dD(t)=dD(-t)), containing only even degrees of t, corresponding to achiral knot projections, and odd polynomials (dD(t)=-dD(-t)), containing only odd degrees of t, which are invariant to a change of orientation. As with every polynomial invariant, the projection polynomial dD(t) sometimes fails to detect isomorphism of knot projections or achirality. For example, for n=10 from 364 non-isomorphic projections of alternating knots it recognizes 363 of them as different, and sometimes fails to detect achirality (that is, sometimes yields an even polynomial d(t) for a chiral knot projection).

The following table contains polynomials dD(t) for all different projections of knots with n 7 crossings. In the first column the symbol of a knot is given in the classic notation, followed by the Conway symbol of the corresponding projection, minimal Dowker code and the projection polynomial dD(t) given by the sequence of its coefficients in descending order.

31     3     4 6 2     1 0    3              
41     2 2     4 6 8  2     1 0   -2   0           
51     5     6 8  10 2  4     1 0    5   0    5        
52     3 2     4 8 10 2  6     1 0   1   0    5        
61     4 2     4 8  12  10  2  6     1 0   -4    -2    3    2     
62     3 1 2     4 8 10  12 2 6     1 0    1   -1   -3   2     
63     2 1 1 2     4  8  10 2 12 6     1 0   2   0    3   0      
71     7     8  10  12  14 2  4  6     1 0    7   0   14   0    7  
72     52      4  10  14  12  2  8 6     1 0   -3   0   -2   0    7  
73     4 3     6 10 12 14 2 4 8     -1 0    -1   0   8   0   7  
74     3 1 3     6  10 12 14 4 2 8     1 0   -4   0   2   0   7  
75     3 2 2     4  10 14  12  2  6  8     -1 0   -2   1   4   0   7  
75'     ((1,3,1),1,1)     4  10  12  14 2 8 6     -1 0    -3    0    4    0    7  
76     2 2 1 2     4  8 12  2  14  6  10     1 0    -1    -1    -1   -2    3  
76'     (((1,2,1),1),1,1)     4 8 12 10 2 14  6     -1 0    0   -1   3   -1    3  
77     2 1 1 1 2     4  8  10  12  2 14 6     1 0    -1    -1   -1   -2   3  
77'     ((((1,2),1),1),1,1)     4 8 12 14 2  6  10     -1 0    0    -1    3    -1   3  

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