Notes for AKT-090917-1/0:11:19

Let [math]\displaystyle{ \mathcal{K} }[/math] denote the space of oriented knots in an oriented [math]\displaystyle{ \mathbb{R}^3 }[/math] and [math]\displaystyle{ A }[/math] be any abelian group. Then, given any invariant [math]\displaystyle{ V: \mathcal{K} \rightarrow A }[/math], we can extend [math]\displaystyle{ V }[/math] to [math]\displaystyle{ 1 }[/math]-singular knots (i.e. knots with one double point) by setting:

[math]\displaystyle{ V^{(1)}(\doublepoint)=V(\overcrossing) - V(\undercrossing) }[/math]

This is analogous to taking the first derivative.