A regular link diagram admits a normalized height realization

Parameterize component $i$ by $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$, and let $U_i=\{u_{i,1},\dots ,u_{i,m_i}\}$ be its nonempty finite set of underpassing parameters. Define $p_i(s)={\prod }_{a=1}^{m_i}(1-\cos (2\pi (s-u_{i,a})))\mathrm{.}$ Every factor is smooth and nonnegative, and it vanishes exactly when $s=u_{i,a}$ in $S^1$. Thus $p_i$ is smooth, vanishes exactly on $U_i$, and is positive somewhere because a finite subset does not exhaust $S^1$. Its maximum $P_i$ is positive by compactness. Set ${\widehat{\eta }}_i(s)=-1+p_i(s)\mathrm{/}{P}_i$. Then $-1\le {\widehat{\eta }}_i\le 0$, equality at $-1$ holds exactly on $U_i$, and all other heights are greater than $-1$.

At every projected double point, the overpassing parameter is not an underpassing parameter: regularity excludes triple points, so the two source parameters at that double point have distinct over/under roles. Hence the new height difference between the over and under branches is strictly positive. Let ${\eta }_i$ be the original height function and choose a smooth $\phi :[0,1]\to [0,1]$ equal to $0$ near $0$ and $1$ near $1$. The family $F_t(i,s)=({\gamma }_i(s),(1-\phi (t)){\eta }_i(s)+\phi (t){\widehat{\eta }}_i(s))$ is smooth and stationary near both endpoints. At each crossing its over-minus- under height is a convex combination of two positive numbers, so it remains positive. Since [F1] lists all projected multiple points, each $F_t$ is injective; its projection remains an immersion, so each compact-source map is a smooth embedding.

The track of the isotopy in step 2.1 is compact, so choose an open ball containing it and apply [F2] with that ball as the support neighborhood. The resulting ambient isotopy of ${\mathbb{R}}^3$ has compact support and therefore extends by the identity at $\mathrm{\infty }$ to an ambient isotopy of $S^3$. It carries the original link to $\widehat{L}$ and fixes the planar projection and over/under data throughout, because every slice of $F$ has those same data. This proves the stated normalized realization.