Planar isotopy of link diagrams

Let $D,D^{\mathrm{\prime }}$ be regular oriented link diagrams (Regular oriented link diagrams), regarded as decorated immersed oriented $4$-valent planar graphs. Then $D$ and $D^{\mathrm{\prime }}$ are planar isotopic when there is an orientation-preserving ambient isotopy $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:{\mathbb{R}}^2\times I\to {\mathbb{R}}^2$, with ${\mathrm{\Phi }}_0=\mathrm{i}\mathrm{d}$ and each ${\mathrm{\Phi }}_t$ an orientation-preserving diffeomorphism (Diffeomorphisms and local diffeomorphisms of manifolds), such that ${\mathrm{\Phi }}_1$ carries the decorated graph of $D$ to that of $D^{\mathrm{\prime }}$: it maps the image curve of $D$ bijectively onto that of $D^{\mathrm{\prime }}$, carries the over/under datum of every double point to the over/under datum of the corresponding double point, and preserves the strand orientations. A planar isotopy of diagrams is a family $D_t={\mathrm{\Phi }}_t(D_0)$ for such a jointly smooth ambient isotopy $\mathrm{\Phi }$, with the decorations transported by ${\mathrm{\Phi }}_t$. The source curves are immersed, not embedded in the plane at their crossings.

Planar isotopies are admitted as moves in the Reidemeister theorem: they carry a regular projection between the exceptional times of a general-position isotopy of the link, and no crossing of the diagram is created or destroyed along them. They are listed separately from the three local moves (Oriented Reidemeister moves), of which R1 and R2 create or destroy crossings, while R3 rearranges three existing crossings and preserves their number. Composing planar isotopies and composing them with the local moves gives the equivalence relation generated by the moves, and every single move below is understood up to planar isotopy outside its small disk.