Notes for AKT-090917-1/0:23:37

Let [math]\displaystyle{ \mathcal{K}_m = \{ m }[/math]-singular knots [math]\displaystyle{ \} }[/math] We have [math]\displaystyle{ V^{(m)}: \mathcal{K}_m \rightarrow A }[/math], since [math]\displaystyle{ V^{(m+1)}=0 }[/math], [math]\displaystyle{ V^{(m)} }[/math] does not distinguish over crossing and under crossings in [math]\displaystyle{ \mathcal{K}_m }[/math]. Let [math]\displaystyle{ \mathcal{D}_m = \mathcal{K}_m / ( \mbox{over crossing}=\mbox{under crossing}) }[/math]. Hence the weight system [math]\displaystyle{ \mathcal{D}_m \rightarrow A }[/math] given by [math]\displaystyle{ W_V = V^{(m)} }[/math] is well-defined.