Notes for AKT-090917-1/0:15:48

Likewise, for an [math]\displaystyle{ m }[/math]-singular knot [math]\displaystyle{ K }[/math], we can define:

[math]\displaystyle{ V^{(m)}(K)=\sum_{K'}{(-1)^{u(K')}V(K')} }[/math]

where the sum is over the [math]\displaystyle{ 2^m }[/math] resolutions of [math]\displaystyle{ K }[/math] and [math]\displaystyle{ u(K') }[/math] is the number of under resolutions made in obtaining [math]\displaystyle{ K' }[/math].

Parallel to the result in calculus that partial derivatives commute, we have that [math]\displaystyle{ V^{(m)}(K) }[/math] is independent of the order in which the double points are resolved.