Oriented links in the three-sphere and ambient isotopy

Work in the smooth category (Smooth manifolds and their smooth charts). Fix once and for all the three-sphere as the one-point compactification

$S^3:={\mathbb{R}}^3\cup \{\mathrm{\infty }\}$

of Euclidean three-space, carrying its standard smooth structure, and fix the standard orientation of $S^3$: the orientation induced by the standard orientation of ${\mathbb{R}}^3$ under the one-point compactification, so that the charts of the standard atlas away from $\mathrm{\infty }$ are orientation-preserving and the orientation is that of a connected oriented $3$-manifold in the sense of Oriented smooth manifolds and oriented charts. For $r>0$ write ${\bar{B}}_r:=\{x\in {\mathbb{R}}^3:\parallel x{\parallel }_2\le r\}$, the closed ball of Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$ with the following restriction convention. Smoothness on a closed ball means local extension to a smooth map on an open neighborhood in Euclidean space; smoothness on $M\times I$, for a smooth manifold $M$ without boundary, means local extension across $t=0,1$ in $M\times \mathbb{R}$. Thus these closed domains are not asserted to be manifolds without boundary in the cited definition.

Links. The standard oriented circle is the circle $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ with its standard counterclockwise orientation, regarded as a smooth $1$-manifold; a finite disjoint union of oriented circles is a smooth manifold $C=S^1\bigsqcup \cdots \bigsqcup S^1$ ($k$ summands, $k\in \mathbb{N}$) diffeomorphic to the disjoint union of $k$ standard oriented circles. An oriented link (with $k$ components) is a smooth embedding

$L\text{\hspace{0.17em}⁣}:C⟶S^3$

of such a disjoint union in the sense of Smooth embeddings. So $L$ is injective, is an immersion, and is a homeomorphism onto its image with the subspace topology, and its image $L(C)$ is a finite disjoint union of smoothly embedded circles. Thus every link considered here is finite and tame: the smoothness of $L$ is exactly the standing convention, and no polygonal or other tame model is used. The components of $L$ are the images of the individual summands, and the number of components is $k$.

Ambient isotopy. Let $I=[0,1]$. An ambient isotopy of $S^3$ is a smooth map

$H\text{\hspace{0.17em}⁣}:S^3\times I⟶S^3$

such that $H_0={\mathrm{i}\mathrm{d}}_{S^3}$, each $H_t\text{\hspace{0.17em}⁣}:S^3\to S^3$, $H_t(x):=H(x,t)$, is a diffeomorphism (Diffeomorphisms and local diffeomorphisms of manifolds), and each $H_t$ is orientation-preserving with respect to the fixed orientation. The last clause is in fact automatic: $t\mapsto H_t$ is continuous in the $C^{\mathrm{\infty }}$ topology of the diffeomorphism group, the orientation sign is locally constant in $t$, and $H_0=\mathrm{i}\mathrm{d}$ is orientation-preserving, so it is kept only for emphasis.

Equivalence of links. Two oriented links $L_0\text{\hspace{0.17em}⁣}:C_0\to S^3$ and $L_1\text{\hspace{0.17em}⁣}:C_1\to S^3$ are equivalent (written $L_0\sim L_1$), or ambient isotopic, when there are an ambient isotopy $H$ and an orientation-preserving diffeomorphism $f:C_0\to C_1$ with $H_1\circ L_0=L_1\circ f$. Thus parametrizations and component labels are irrelevant, while the induced orientation of each image component is preserved. The relation is an equivalence relation on links: it is reflexive with $H=\mathrm{i}\mathrm{d}$, transitive by composing isotopies, and symmetric because a smooth family of diffeomorphisms has a smooth family of inverses. An equivalence class is an (oriented) link type.

Moving off $\mathrm{\infty }$. Because the definition of equivalence allows the isotopy to move a link through $\mathrm{\infty }$, the standard practice of this page is to first apply an ambient isotopy that carries a given link into the open ball, and only then to project; every projection and every Reidemeister move below is read after moving the link off $\mathrm{\infty }$ into ${\mathbb{R}}^3$ in this way. The orientation-preserving clause in the definition of ambient isotopy is exactly what the oriented Reidemeister moves and the oriented closure construction must respect. The unoriented theory uses the same ambient isotopies but forgets the orientations of the link components.

Links in the three-sphere versus links in ${\mathbb{R}}^3$. A link contained in ${\mathbb{R}}^3\subset S^3$ is a link in $S^3$; conversely every link may be assumed after isotopy to lie in ${\mathbb{R}}^3$, by choosing a point outside the compact one-dimensional image and moving that point to $\mathrm{\infty }$ by a smooth rotation of the standard sphere. When comparing links already in ${\mathbb{R}}^3$, an isotopy can also be arranged to avoid $\mathrm{\infty }$ throughout; the track-avoidance argument is given in Reidemeister's theorem for oriented diagrams.