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

Likewise, for an $m$-singular knot $K$, we can define:

$V^{(m)}(K)=\sum _{K'}{(-1)^{u(K')}V(K')}$

where the sum is over the $2^{m}$ resolutions of $K$ and $u(K')$ is the number of under resolutions made in obtaining $K'$.

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