Notes for AKT-090917-1/0:20:08

Definition: A knot invariant [math]\displaystyle{ V }[/math] is of Vassiliev type [math]\displaystyle{ m }[/math] if [math]\displaystyle{ V^{(m+1)} = 0 }[/math] (on the whole space of [math]\displaystyle{ (m+1) }[/math]-singular knots).

Notation: We drop the superscript in [math]\displaystyle{ V^{(m)} }[/math] since for each [math]\displaystyle{ m }[/math], [math]\displaystyle{ V^{(m)} }[/math] is only defined for [math]\displaystyle{ m }[/math]-singular knots.

We can also express the 'type [math]\displaystyle{ m }[/math]' condition as:

[math]\displaystyle{ V(\doublepoint ... \doublepoint)=0 }[/math]

whenever we have more than [math]\displaystyle{ m }[/math] double points.