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

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

Failed to parse (unknown function "\doublepoint"): {\displaystyle V^{(1)}(\doublepoint)=V(\overcrossing) - V(\undercrossing)}

This is analogous to taking the first derivative.