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

Let [math]\displaystyle{ \mathcal{K}_m = \{ m }[/math]-singular knots [math]\displaystyle{ \} }[/math]

Given [math]\displaystyle{ V }[/math] of type [math]\displaystyle{ m }[/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.