ScalingStacks

Lemma 4.1. Let $L\subset B$ be a link embedded in a 3-ball. Let $\phi_{0}$ and $\phi_{1}$ be two diffeomorphisms from $B$ to ${\mathbb{R}}^{3}$ such that $\phi_{0}(L)$ and $\phi_{1}(L)$ are generic. Then we have the following:

1. (1)

   There exists a continuous family of diffeomorphisms $\phi_{t}$ for $t\in{[0,1]}$, such that $\phi_{t}(L)$ is generic for all but finitely many $t\in{[0,1]}$, at which a Reidemeister move occurs or the crossing height order changes.

2. (2)

   Given two such families $\phi_{t,0}$ and $\phi_{t,1}$, both interpolating between $\phi_{0}$ and $\phi_{1}$, then there exists a continuous family $\phi_{t,s}$ of diffeomorphisms interpolating between the families $\phi_{t,0}$ and $\phi_{t,1}$, for which the parameter space ${[0,1]}\times{[0,1]}$ is stratified such that:

   • •

     $\phi_{t,s}(L)$ is generic for $(t,s)$ in any codimension-0 stratum,

   • •

     $\phi_{t,s}(L)$ undergoes a Reidemeister move or the crossing height order changes as $(t,s)$ crosses through a codimension-1 stratum,

   • •

     $\phi_{t,s}(L)$ has a movie move as monodromy if $(t,s)$ loops around a codimension-2 stratum.