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An embedding identifies one mathematical structure with a structure-preserving copy inside another. In topology it is a homeomorphism onto its image with the subspace topology; an algebraic embedding is an injective homomorphism. A smooth embedding additionally has injective differential everywhere, so its image is an embedded submanifold.

Suppose a compact skeleton $Y$ embeds as $e(Y)\subset {\mathbb{R}}^m$. Attach a disk by $a:S^{k-1}\to Y$ and extend its boundary coordinate map radially by $G(ru)=re(a(u))$. Map the disk to $(G(x),(1-|x|)x,1-|x|)$ and $Y$ to $(e(y),0,0)$. The boundary identifications agree; a positive last coordinate recovers each interior point. This defines a continuous injection of the compact quotient into Euclidean space, hence an embedding. Repeating proves Hausdorffness and Euclidean embeddability for every finite CW complex, without assuming triangulability of arbitrary attaching maps.

Knot theory studies closed curves in three-dimensional Euclidean space that cannot intersect themselves, and asks which can be deformed into one another without cutting or passing one strand through another. Its basic object is a knot: a smooth embedding of a circle, considered up to ambient isotopy. Several disjoint knots form a link. More generally, knot theory studies such embeddings in other manifolds and in higher dimensions.

The central surprise is that a flexible circle can carry permanent information about how it sits in space. A trefoil knot and an unknot each consist of just one circle, but no ambient isotopy turns one into the other. A knot diagram translates this spatial question into a finite drawing; Reidemeister moves describe the changes of drawing that preserve the knot. Knot invariants translate it into computations: unequal Alexander polynomials of a knot, Jones polynomials, or knot groups certify different knots. Equal values of a particular knot invariant generally leave the question open; even an isomorphism of knot groups alone is not a complete classification.

An early foundation for working with knot diagrams is Kurt Reidemeister's 1927 paper Elementare Begründung der Knotentheorie, published in volume 5 of Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, pages 24–32.^[ref] Marc Lackenby's authored notes Elementary knot theory, Section 2.4, explain the Reidemeister move theorem and its role in searching for a certificate that two knot diagrams represent the same knot.^[ref]

Orientability asks whether local choices of handedness can agree everywhere on a manifold. On a smooth manifold, an orientation of a smooth manifold chooses an orientation of a vector space for each tangent space, continuously with position. An ordered basis describes the same handedness as another one when the change-of-basis determinant is positive. An oriented atlas implements this condition through positive Jacobian determinants of the smooth transition maps.

A paper strip makes the obstruction visible. Join its short ends without twisting: the resulting untwisted band admits a consistent choice of unit normal. Add a half-turn before joining: on the Möbius band, following the core circle of a Möbius band reverses that choice when you first return to the same point. A local decision about handedness has encountered a global contradiction. The normal reversal around the Möbius core makes this precise for an explicit model.

Both rectangles have coordinates $(u,v)\in [0,1]\times [-1,1]$. The untwisted band identifies $(0,v)$ with $(1,v)$; the Möbius band identifies $(0,v)$ with $(1,-v)$. Across a seam the corresponding local coordinate changes preserve or reverse $v$, giving determinants $+1$ and $-1$. This sign explains the handedness change; the drawing also tracks the two boundary circles of the untwisted band and the single boundary circle of the Möbius band.

For a smooth topological surface embedded in ${\mathbb{R}}^3$, choosing an orientation of a surface is equivalent to choosing a continuous unit normal. This picture is specific to surfaces in three-dimensional space: the general definition uses compatible local orientations, without requiring an embedding. Orientability is a topological invariant, so a homeomorphism cannot turn a Möbius band into a untwisted band. The sphere and torus are orientable surfaces; the Klein bottle is not. ^[ref]

The distinction matters when a sign has geometric meaning. A signed flux surface integral needs an orientation of a surface, and Stokes theorem relates that choice to the outward-normal-first boundary orientation. Ordinary unsigned surface area does not require orientability. Thus a Möbius band still has a perfectly meaningful area even though it has no continuous global unit normal.

A connected orientable manifold of positive dimension has two possible orientations: reversing every local choice gives the other one. If a manifold has several connected components, each component can be oriented independently when it is orientable. For topological surfaces, a useful obstruction criterion is the presence of a Möbius band: a surface is nonorientable exactly when it contains such a band. ^[ref]

A subset $X\subset M^m$ is a $d$-dimensional embedded submanifold if, with the subspace topology, every point has a smooth ambient chart in which $X$ is the intersection of the coordinate domain with ${\mathbb{R}}^d\times \{0\}$. These slice charts give its smooth structure, and its inclusion is an embedding.

The inverse function theorem says that a smooth map $F:U\subset {\mathbb{R}}^m\to {\mathbb{R}}^m$ with invertible derivative at $a$ restricts to a diffeomorphism between open neighborhoods of $a$ and $F(a)$. A point $q\in N$ is a regular value of a smooth map $f:M^m\to N^n$ when $d{f}_x:T_{x}M\to T_{q}N$ is surjective at every $x\in f^{-1}(q)$.

Choose local coordinates centered at $x$ and $q$. At a fiber point, rank $n$ means that, after relabeling coordinates, the first $n$ columns of the coordinate derivative form an invertible matrix. DefineIts derivative is block upper triangular with that invertible block and an identity block, hence invertible. The inverse function theorem makes $F$ a new ambient coordinate chart in which $f^{-1}(q)$ is exactly $v_1=\cdots =v_n=0$. Therefore the regular level set theorem giveswhen the fiber is nonempty. An empty fiber is regular vacuously; if $m<n$, it is the only possible regular fiber. The construction does not assign a negative dimension to a nonempty manifold.

For the projective hyperplane, the mapis injective with inverse $[x_0:x_1:0]\mapsto [x_0:x_1]$ on its image. At every image point, either $x_0$ or $x_1$ is nonzero; the corresponding standard projective chart uses $x_2/x_i$ as one coordinate and cuts the image out by setting it to zero. The map and inverse are smooth in these charts, and the slice condition provesNevertheless, ${\mathbb{RP}}^2\setminus X$ is the single affine chart $x_2\ne 0$, diffeomorphic to ${\mathbb{R}}^2$, so it is connected. If a smooth real function had exactly this zero set with zero regular, the inverse function theorem near any point of $X$ would give both positive and negative values off $X$. On the connected complement it never vanishes, so continuity requires its sign to be constant. This contradiction is the obstruction that regular zero hypersurfaces have disconnected complements. ThusIt illustrates why local defining equations for an embedded hypersurface need not glue to a global real defining function, as in real projective hyperplane is not a global regular zero set.

A smooth embedding is an injective immersion that is a homeomorphism onto its image with the subspace topology.

This is the differentiable specialization of an embedding, with an injective differential in addition to its topological embedding property.