The Jones Polynomial

The knots 6_1 and 9_46 have the same Alexander polynomial. Their Jones polynomials are different, though:

[math]\displaystyle{ In[1]:=\lt /tt\gt \lt code\gt Jones[Knot[6, 2]][q] \lt /code\gt Out[1]=\lt /tt\gt \lt math\gt q-1+\frac{2}{q}-\frac{2}{q^2}+\frac{2}{q^3}-\frac{2}{q^4}+\frac{1}{q^5} }[/math]

Again:

In[1]:= Jones[Knot[6, 1]][q]

Out[1]=

[math]\displaystyle{ q^2-q+2-\frac{2}{q}+\frac{1}{q^2}-\frac{1}{q^3}+\frac{1}{q^4} }[/math]

[math]\displaystyle{ In[2]:=\lt /tt\gt \lt code\gt Jones[Knot[9, 46]][q] \lt /code\gt Out[2]=\lt /tt\gt \lt math\gt 2-\frac{1}{q}+\frac{1}{q^2}-\frac{2}{q^3}+\frac{1}{q^4}-\frac{1}{q^5}+\frac{1}{q^6} }[/math]

Again:

In[2]:= Jones[Knot[6, 1]][q]

Out[2]=

[math]\displaystyle{ 2-\frac{1}{q}+\frac{1}{q^2}-\frac{2}{q^3}+\frac{1}{q^4}-\frac{1}{q^5}+\frac{1}{q^6} }[/math]