Notes for AKT-090917-2/0:27:14

Using the skein relation, we see that [math]\displaystyle{ J(K_1) }[/math] has a factor of [math]\displaystyle{ x }[/math] where [math]\displaystyle{ K_1 }[/math] is a knot with one double point. Analogously, for [math]\displaystyle{ K_{n+1} }[/math] - a knot with [math]\displaystyle{ k+1 }[/math] double points, [math]\displaystyle{ J(K_{n+1}) }[/math] will have a factor of [math]\displaystyle{ x^{n+1} }[/math]. In particular, the coefficient of [math]\displaystyle{ x^n }[/math], [math]\displaystyle{ J_n }[/math], will be zero.