Notes for AKT-090917-2/0:20:43

Proof of the theorem from last hour: [math]\displaystyle{ j_{(n, \cdot)} }[/math] is a type [math]\displaystyle{ n }[/math] invariant. (in fact we may prove the same for both the Conway and HOMFLY-PT polynomial) Substitute [math]\displaystyle{ q=e^x }[/math] into the Jones skein relation, simplify [math]\displaystyle{ J( }[/math] double point [math]\displaystyle{ ) }[/math] and deduce the value of [math]\displaystyle{ J }[/math] on a [math]\displaystyle{ n+1 }[/math] singular knot [math]\displaystyle{ \Rightarrow j_n }[/math] vanishes on [math]\displaystyle{ \mathcal{K}_{n+1} }[/math].