Oriented crossing signs, diagram writhe, and blackboard framing

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the smooth normal-bundle structure used in the framing clause. For a knot component $K$, its normal bundle is the fibrewise quotient $\nu K=T{S}^3{\mathrm{\mid }}_K\mathrm{/}TK$, and a normal section is a smooth section of this bundle.

Let $D$ be a regular oriented link diagram in a projection plane $P$ of an oriented $3$-sphere, after moving the link off the point at infinity. Fix the unit viewing normal $n$ pointing toward the viewer. Orient $P$ by the rule that an ordered basis $(u,v)$ of $P$ is positive exactly when $(n,u,v)$ is a positive basis of the ambient $3$-space. At a crossing $c$, let $v_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}}$ and $v_{\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}}$ be the nonzero projected tangent vectors following the orientations of the overpassing and underpassing strands. Define $\mathrm{sgn}(c):=\mathrm{sgn}{\det }_P(v_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}},v_{\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}})\in \{+1,-1\}\mathrm{.}$ Reversing the orientation of $P$ reverses every crossing sign.

This convention makes the positive geometric Artin half twist positive. In the braid chart $(x,y,t)$ with ambient orientation $(x,y,t)$, view along $+y$; the induced positive orientation of the $(x,t)$ projection plane is the ordered basis $({\mathrm{\partial }}_t,{\mathrm{\partial }}_x)$. For the positive half twist, label $i$ moves from $x=-h$ to $x=h$ below the midpoint, while label $i+1$ moves from $x=h$ to $x=-h$ above it, with $h>0$. Thus the over and under projected tangent vectors in coordinates $(t,x)$ are $(1,-2h)$ and $(1,2h)$, whose determinant is $4h>0$.

The writhe of $D$ is $\mathrm{wr}(D):={\sum }_{c\text{ a crossing of }D}\mathrm{sgn}(c)\mathrm{.}$ For a link component $K_i$, its self-writhe ${\mathrm{wr}}_i(D)$ is the sum over crossings whose two branches belong to $K_i$; the total writhe also includes all mixed crossings between distinct components.

The blackboard framing of a component of $D$ is the homotopy class of the normal section represented by the positive quarter-turn in $P$ of the projected oriented tangent. The projection is an immersion, so this vector never vanishes; it is perpendicular to the projected tangent, hence also to the spatial tangent, and therefore represents a nonzero normal section of the component in the ambient $3$-sphere.

In the displayed local chart, the over tangent is the positive horizontal vector and the under tangent is the positive vertical vector, so the determinant is $+1$. This figure illustrates the determinant rule only; no proof below depends on a picture.

Smooth oriented links and ambient orientation

Regular diagrams and over/under data

Positive geometric Artin half twist

Normal bundle quotient

Smooth normal-bundle structure

Countable-choice hypothesis for the smooth normal bundle