The Jones Polynomial
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(For In[1] see Setup)
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In Naming and Enumeration we checked that the knots 6_1 and 9_46 have the same Alexander polynomial. Their Jones polynomials are different, though:
In[3]:=
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Jones[Knot[6, 1]][q]
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Out[3]=
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-4 -3 -2 2 2
2 q - q q - - - q q
q
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In[4]:=
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Jones[Knot[9, 46]][q]
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Out[4]=
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-6 -5 -4 2 -2 1
2 q - q q - -- q - -
3 q
q
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L8a6 |
On links with an even number of components the Jones polynomial is a function of , and hence it is often more convenient to view it as a function of , where :
In[5]:=
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Jones[Link[8, Alternating, 6]][q]
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Out[5]=
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-(9/2) -(7/2) 3 3 4 3/2
-q q - ---- ---- - ------- 3 Sqrt[q] - 2 q
5/2 3/2 Sqrt[q]
q q
5/2 7/2
2 q - q
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In[6]:=
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PowerExpand[Jones[Link[8, Alternating, 6]][t^2]]
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Out[6]=
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-9 -7 3 3 4 3 5 7
-t t - -- -- - - 3 t - 2 t 2 t - t
5 3 t
t t
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The Jones polynomial attains