View source for Notes for Aarhus-130607-1/0:17:09

But I didn't say it precisely enough - I said it wrong. The correct statement is that if two knots $\gamma_1$ and $\gamma_2$ embedded on surfaces $\Sigma_1$ and $\Sigma_2$ are equivalent as virtual knots, and if $\Sigma_1$ and $\Sigma_2$ are both of genus $g$ where $g$ is the minimal genus on which $\gamma_{1,2}$ may be embedded, then there is a diffeomorphism $f\colon\Sigma_1\to\Sigma_2$ such that $f(\gamma_1)$ is isotopic to $\gamma_2$.