Framed knots, self-linking, and the Seifert framing

Assume the Axiom of Countable Choice. For an oriented smooth knot $K:S^1\hookrightarrow S^3$, its normal bundle is $\nu K=T{S}^3{\mathrm{\mid }}_K\mathrm{/}TK$, oriented so that a positive tangent to $K$ followed by a positive basis of a normal fibre gives the ambient orientation. A framing of $K$ is a homotopy class through smooth nowhere-zero sections of $\nu K$. The normal bundle is made a smooth vector bundle under this axiom; a framed knot includes the section class as data, not a choice made afterwards. For the relative integer use the round metric transferred from the unit sphere in ${\mathbb{R}}^4$ by the explicit diffeomorphism $x⟼(\frac{2x}{1+\parallel x{\parallel }^2},\frac{\parallel x{\parallel }^2-1}{1+\parallel x{\parallel }^2}),\mathrm{\infty }⟼(0,0,0,1)\mathrm{.}$ It induces a smooth bundle metric on $\nu K$. The resulting winding number is independent of the metric: any two smooth fibre metrics interpolate through positive metrics and the corresponding angle maps are homotopic.

Given a reference framing $[s_0]$ and another framing $[s_1]$, use a bundle metric and the positive quarter-turn $J$ on each oriented normal plane. Choose representatives $s_0,s_1$ and normalize both to unit sections ${\widehat{s}}_i(x):=s_i(x)\mathrm{/}\parallel s_i(x)\parallel$. Write ${\widehat{s}}_1(x)=\cos \theta (x){\widehat{s}}_0(x)+\sin \theta (x)J{\widehat{s}}_0(x)\mathrm{.}$ The resulting map $S^1\to S^1$ has a continuous angle lift along a parameter interval $[0,1]$: subdivide into finitely many intervals whose images lie in open semicircles and choose consecutive local arguments agreeing at endpoints. Since the map is closed, the lift satisfies $\theta (1)-\theta (0)=2\pi m$ for some $m\in \mathbb{Z}$; different lifts differ by a constant multiple of $2\pi$, and homotopies keep $m$ fixed. Conversely, the difference between $\theta (t)$ and $\theta (0)+2\pi mt$ is periodic, so linear interpolation of these angles gives a homotopy to the degree-$m$ rotation of ${\widehat{s}}_0$. Every integer occurs by such a rotation. Thus $m$ classifies the framing class relative to $[s_0]$ and gives the relative framing integer. Once a reference framing is supplied, the framing classes are an integer torsor; no absolute integer is assigned before choosing a reference.

The self-linking number of a framed knot $(K,[s])$ is the linking number of $K$ with a sufficiently small push-off along $s$, oriented parallel to $K$. The push-off depends on the representative section, small radius, and normalized tubular chart, but its linking integer is independent of these choices by the preceding well-definedness result.

If a compact oriented surface $\mathrm{\Sigma }$ is supplied with boundary $K$, its Seifert framing relative to $\mathrm{\Sigma }$ is the homotopy class of the inward unit conormal to $K$ in $\mathrm{\Sigma }$, viewed as a nonzero normal section to $K$ in $S^3$. This definition is relative to the supplied surface; it neither asserts that every knot bounds such a surface nor asserts the self-linking-zero result.

Smooth oriented knots in the three-sphere

Countable-choice hypothesis for smooth normal bundles

Normal bundle quotient

Smooth structure on the normal bundle

Smooth metric on each normal fibre

Well-definedness of small framed push-offs

Diagram linking number used for self-linking

For a link component, self-linking is computed in the two-component sublink formed by that component and its push-off; crossings with other link components do not enter. The total writhe of a link remains the signed sum over all crossings and is a separate quantity used for polynomial normalizations.