Oriented Links, Braid Closures, and Markov Equivalence

Compactness of the image and a relatively compact neighbourhood. If $M=\mathrm{\varnothing }$, take $H_t={\mathrm{i}\mathrm{d}}_N$ for every $t$; all extension and support assertions are then immediate. Assume $M\mathrm{\ne }\mathrm{\varnothing }$. The set $K:=F(M\times I)\subseteq N$ is compact, being the continuous image of the compact space $M\times I$, and it is closed in $N$ because smooth manifolds are Hausdorff. Each point of $K$ has a coordinate ball whose closure is compact and contained in $W$; finitely many of these balls cover $K$, and their union $V$ is an open neighbourhood of $K$ with compact closure $\bar{V}\subseteq W$. Fix such a $V$.

The velocity field along the slices, as a field along a graph. Extend $F$ to $\mathbb{R}\times M$ by $F_t=F_0$ for $t<0$ and $F_t=F_1$ for $t>1$. This extension is smooth because the given $F$ is constant on the full endpoint collars of width $\epsilon$, and every extended slice remains an embedding. Put $J:=\mathbb{R}$ and consider the map $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:J\times M\to J\times N$, $\mathrm{\Theta }(t,x):=(t,F_t(x))$, with image $S:=\mathrm{\Theta }(J\times M)$. $\mathrm{\Theta }$ is injective because its first coordinate is $t$; its derivative at $(t,x)$ equals $(\mathrm{d}t,{\mathrm{\partial }}_{t}F(x,t)\mathrm{d}t+\mathrm{d}(F_t{)}_x)$, which is injective because $\mathrm{d}(F_t{)}_x$ is injective by [F6]; and $\mathrm{\Theta }$ is proper: for a compact subset $L\subseteq J\times N$, its time projection is compact, and ${\mathrm{\Theta }}^{-1}(L)$ is closed in the compact product of that projection with $M$. The inverse on the image is continuous locally by the embedding property of its slices, or globally by this properness. Its injective derivative therefore makes $S$ a closed embedded submanifold without boundary of $J\times N$, of dimension $1+\dim M$. The constant time extension avoids applying a boundaryless extension theorem to a graph with boundary. The assignment $Y(t,F_t(x)):=(0,{\mathrm{\partial }}_{t}F(x,t))\in T_{(t,F_t(x))}(J\times N)$ is well defined because each $F_t$ is injective, and it is a smooth vector field along $S$: near a point of $S$ the inverse $y\mapsto x$ of $F_t$ is smooth by [F6], so $Y$ is the composite of smooth maps; along $S$ it is everywhere tangent to the splitting of $T(J\times N)$ into the $J$-direction and $TN$.

Extension and truncation. By [F4] applied to the closed embedded submanifold $S$ of the smooth manifold $J\times N$, the field $Y$ extends to a global smooth vector field $\tilde{Y}$ on $J\times N$ with $\tilde{Y}{\mathrm{\mid }}_S=Y$. Write $\tilde{Y}=(a,X)$ in the splitting $T(J\times N)\cong \mathbb{R}\oplus TN$, so that $X$ is a smooth family $X_t:=X(t,\cdot )$ of vector fields on $N$ with $X_t(F_t(x))={\mathrm{\partial }}_{t}F(x,t)$ for all $x\in M$ and $t\in J$. Choose a smooth function $\psi \text{\hspace{0.17em}⁣}:N\to [0,1]$ with $\psi =1$ on a neighbourhood of $K$ and $\mathrm{supp}\psi \subseteq V$: the open sets $V$ and $N\setminus K$ cover $N$ because $K$ is closed, so [F2] applied to this two-element cover produces $\psi$ as the member subordinate to $V$, whose support lies in $V$. The other member has closed support contained in $N\setminus K$; its support complement is an open neighbourhood of $K$ on which that other member vanishes and hence $\psi =1$. Choose a smooth $\beta \text{\hspace{0.17em}⁣}:\mathbb{R}\to [0,1]$ with $\beta =1$ on $[\epsilon ,1-\epsilon ]$ and $\beta =0$ on $(-\mathrm{\infty },\frac{\epsilon }{2}]\cup [1-\frac{\epsilon }{2},\mathrm{\infty })$, and put $X_t^{\mathrm{\prime }}:=\beta (t)\psi X_t$ for $t\in J$. Every $X_t^{\mathrm{\prime }}$ is a smooth vector field on $N$ with $\mathrm{supp}{X}_t^{\mathrm{\prime }}\subseteq \bar{V}$, so ${\bigcup }_{t\in I}\mathrm{supp}{X}_t^{\mathrm{\prime }}\subseteq \bar{V}$ is compact.

The ambient isotopy. Apply [F5] on the compact interval $I$ to the family $X_t^{\mathrm{\prime }}$ of step 2.1 and let ${\mathrm{\Psi }}_{t,s}$ be the resulting global evolution operator. Put $H_t:={\mathrm{\Psi }}_{t,0}$ for $t\in I$. Then $H_0={\mathrm{i}\mathrm{d}}_N$, the map $H\text{\hspace{0.17em}⁣}:N\times I\to N$ is smooth, and each $H_t$ is a diffeomorphism with inverse $H_t^{-1}={\mathrm{\Psi }}_{0,t}$, by the cocycle law in [F5]. Since $\mathrm{supp}{X}_t^{\mathrm{\prime }}\subseteq V$ for every $t$, every trajectory of $X^{\mathrm{\prime }}$ starting outside $V$ is constant, so $H_t={\mathrm{i}\mathrm{d}}_N$ on $N\setminus V$ for every $t$; a fortiori $H_t$ is the identity outside $W$, because $V\subseteq W$. Finally $X_t^{\mathrm{\prime }}=0$ whenever $t\le \frac{\epsilon }{2}$ or $t\ge 1-\frac{\epsilon }{2}$, because $\beta$ vanishes there. Uniqueness of the evolution with zero velocity gives ${\mathrm{\Psi }}_{t,s}={\mathrm{i}\mathrm{d}}_N$ when $s,t$ belong to either one of those intervals. Starting at time zero therefore gives $H_t={\mathrm{i}\mathrm{d}}_N$ on the initial interval. On the terminal interval the cocycle law gives $H_1={\mathrm{\Psi }}_{1,t}\circ H_t=H_t$, so the ambient isotopy is stationary at its final map there.

The isotopy identity. Fix $x\in M$ and consider $\gamma (t):=F_t(x)$. Then $\gamma (0)=F_0(x)$, and for every $t\in I$ the derivative is ${\gamma }^{\mathrm{\prime }}(t)={\mathrm{\partial }}_{t}F(x,t)$. Since $\gamma (t)\in K$ for all $t$, we have $\psi (\gamma (t))=1$, and therefore $X_t^{\mathrm{\prime }}(\gamma (t))=\beta (t){\mathrm{\partial }}_{t}F(x,t)\mathrm{.}$ If $t\in [\epsilon ,1-\epsilon ]$ then $\beta (t)=1$ and this equals ${\gamma }^{\mathrm{\prime }}(t)$; if $t\mathrm{\notin }[\epsilon ,1-\epsilon ]$ then either $\beta (t)=0$ or, by the standing hypothesis that $F$ is constant near the ends, ${\mathrm{\partial }}_{t}F(x,t)=0$; in both cases $X_t^{\mathrm{\prime }}(\gamma (t))=0={\gamma }^{\mathrm{\prime }}(t)$. Hence $\gamma$ solves the initial-value problem $\dot{\gamma }=X_t^{\mathrm{\prime }}(\gamma )$, $\gamma (0)=F_0(x)$, and by the defining property of the evolution operator in [F5] the unique solution is $t\mapsto {\mathrm{\Psi }}_{t,0}(F_0(x))=H_t(F_0(x))$.

Conclusion. Step 4.1 gives $H_t\circ F_0=F_t$ for every $t\in I$; step 3.1 gives that $H_0={\mathrm{i}\mathrm{d}}_N$, that every $H_t$ is a diffeomorphism, that $H_t={\mathrm{i}\mathrm{d}}_N$ outside $W$, and that $H_t={\mathrm{i}\mathrm{d}}_N$ for $t\in [0,\frac{\epsilon }{2}]$ and $H_t=H_1$ for $t\in [1-\frac{\epsilon }{2},1]$. Thus $H$ is an ambient isotopy of $N$ extending the isotopy $F$ of $F_0(M)$ and supported in the prescribed neighbourhood $W$.

Empty, singleton and uniform margins. For $n=0$ choose the empty smooth braid and constant empty isotopy; every assertion is vacuous. Henceforth $n\ge 1$. The finite disk-boundary function $t\mapsto {\min }_j(1-\mathrm{\mid }{z}_j(t)\mathrm{\mid })$ is positive and continuous on $I$, so has positive minimum. If $n\ge 2$, the finite collision function $t\mapsto {\min }_{i<j}\mathrm{\mid }{z}_i(t)-z_j(t)\mathrm{\mid }$ also has positive minimum. Choose $m>0$ at most both minima when $n\ge 2$, and at most the boundary minimum alone when $n=1$. Put $\eta =\min \{\rho \mathrm{/}4,m\mathrm{/}8\}$, so $3\eta <\rho$ and $3\eta <m\mathrm{/}2$. No minimum of an empty pair list is used.

Approximate on a boundaryless domain. Extend the continuous tuple $z:I\to {\mathbb{R}}^{2n}$ to $\mathbb{R}$ by the constant tuple $z(0)$ for $t<0$ and $z(1)$ for $t>1$. This extension is continuous because its values agree at $0,1$. Apply [F2] on the boundaryless smooth manifold $\mathbb{R}$, with constant error $\eta$, and restrict the resulting smooth map to $I$. It gives $g=(g_1,\dots ,g_n)$ with $\mathrm{\mid }{g}_j(t)-z_j(t)\mathrm{\mid }<\eta$ for every $j,t$. Thus no boundaryless Whitney assertion is applied directly to $I$.

Exact endpoint collars, including preserved smooth germs. Choose $\delta \in (0,1\mathrm{/}2)$ so $\mathrm{\mid }{z}_j(t)-q_j\mathrm{\mid }<\eta$ on $[0,\delta ]$ and $\mathrm{\mid }{z}_j(t)-z_j(1)\mathrm{\mid }<\eta$ on $[1-\delta ,1]$ for all $j$. Take smooth cutoffs ${\chi }_0,{\chi }_1$ supported in these disjoint collars and equal to one on the half-sized endpoint collars. Ordinarily use the constant collar data $a_{j0}(t)=q_j$, $a_{j1}(t)=z_j(1)$. If the original strands are smooth in an endpoint neighbourhood, shrink the corresponding collar into that neighbourhood and instead use $a_{j0}(t)=z_j(t)$ or $a_{j1}(t)=z_j(t)$ there. The products with their cutoffs extend smoothly by zero off the collars. Define $h_j=(1-{\chi }_0-{\chi }_1)g_j+{\chi }_0{a}_{j0}+{\chi }_1{a}_{j1}$. This is smooth with $h_j(0)=q_j$, $h_j(1)=z_j(1)$. In the already-smooth case it agrees with $z_j$ throughout the smaller original endpoint neighbourhood, rather than replacing that neighbourhood by a constant.

All collar choices satisfy the same error bound. On either collar, the replacement error $\mathrm{\mid }{a}_{j\mathrm{\ell }}-z_j\mathrm{\mid }$ is less than $\eta$ for constant data and zero for preserved original data. Since the cutoffs have disjoint supports and weights sum to one, $\mathrm{\mid }{h}_j-z_j\mathrm{\mid }\le (1-{\chi }_0-{\chi }_1)\mathrm{\mid }{g}_j-z_j\mathrm{\mid }+{\chi }_0\mathrm{\mid }{a}_{j0}-z_j\mathrm{\mid }+{\chi }_1\mathrm{\mid }{a}_{j1}-z_j\mathrm{\mid }<3\eta \le 3m\mathrm{/}8<m\mathrm{/}2$. This covers both arbitrary continuous endpoints and already-smooth endpoint neighbourhoods.

The repaired motion is a braid, with the same endpoints. For all $i\mathrm{\ne }j$ and all $t$, $\mid h_i(t)-h_j(t)\mid \ge \mid z_i(t)-z_j(t)\mid -\mid h_i(t)-z_i(t)\mid -\mid h_j(t)-z_j(t)\mid >m-\frac{3m}{8}-\frac{3m}{8}=\frac{m}{4}>0,$ so the values $h_1(t),\dots ,h_n(t)$ are pairwise distinct; and $\mid h_j(t)\mid \le \mid z_j(t)\mid +\frac{3m}{8}<1$, so they lie in $D^{\circ }$. Together with $h_j(0)=q_j$ and $\{h_j(1)\}=\{z_j(1)\}=\{q_j\}$ from [F3] and step 3.1, this shows that $h$ is a braid based at $Q$, smooth by step 3.1, whose endpoint permutation equals that of $\beta$ because $h_j(1)=z_j(1)$ for every $j$.

The straight-line isotopy. Define $Z_j(s,t):=(1-s)z_j(t)+s{h}_j(t)$ for $(s,t)\in I\times I$. It is jointly continuous, $Z(0,\cdot )=\beta$ and $Z(1,\cdot )=h$, and every slice is a braid based at $Q$: for all $s,t,i\mathrm{\ne }j$, $\mid Z_j(s,t)-z_j(t)\mid =s\mid h_j(t)-z_j(t)\mid <\frac{3m}{8},\mid Z_j(s,t)\mid \le \mid z_j(t)\mid +\frac{3m}{8}<1,$ so the $n$ values are pairwise distinct (their pairwise distances are at least $m-\frac{3m}{8}-\frac{3m}{8}=\frac{m}{4}>0$) and lie in $D^{\circ }$; moreover $Z_j(s,0)=(1-s)q_j+s{q}_j=q_j$ and $Z_j(s,1)=(1-s)z_j(1)+s{z}_j(1)=z_j(1)$ for every $s$. Hence $Z$ is a braid isotopy from $\beta$ to the smooth braid $h$.

Conclusion. By step 6.1 the smooth braid $h$ is braid-isotopic to $\beta$ through the isotopy $Z$, whose deviation from $\beta$ satisfies $\mid Z_j(s,t)-z_j(t)\mid =s\mid h_j(t)-z_j(t)\mid <3\eta <\rho \text{for all }j,s,t,$ by the choice of $\eta$ in step 1.1. The isotopy keeps every bottom endpoint fixed pointwise and every individual top endpoint fixed, so the endpoint permutation is preserved by step 5.1. Where a smooth original collar was retained in step 3.1, $h=z$ there, so the entire straight-line isotopy is also fixed there. Taking ${\beta }^{\mathrm{\prime }}:=h$ proves the statement for the prescribed $\rho$, and $\rho >0$ was arbitrary.

Smooth braid isotopies give isotopies of closures. Assume the family is smooth and its disk-coordinate strands are constant on collars of height $0,1$. The endpoint permutation $\pi$ is independent of the family parameter by [F4]. For each $\pi$-cycle of length $k$, concatenate its $k$ strands on $\mathbb{R}\mathrm{/}k\mathbb{Z}$, exactly as in [F3]. Their fixed endpoint collars make all jets agree at the seams. Thus the source is the compact one-dimensional manifold $M={⨆}_{\text{cycles of }\pi }\mathbb{R}\mathrm{/}k\mathbb{Z}$, and the concatenations followed by the fixed $\phi$ give a smooth family $F_u:M\to S^3\setminus A$. Each $F_u$ is an embedding by the distinct-points condition and the component argument of [F3], with its orientation inherited from the increasing cycle parameter. Reparametrize $u$ to make the family constant near $0,1$. By [F2] this isotopy extends to an ambient isotopy of $S^3$, supported away from $A$ in a neighbourhood of its compact image. Each image still has exactly $n$ intersections with every page. If a literal smooth closure has matching cycle-seam jets but is not constant in endpoint collars, interpolate its common height map from the identity to the fixed flat map of [F3]. The matching jets remain matching in every smooth parameter slice, so this gives another compact cycle embedding family and [F2] identifies that literal closure with its constant-collar model.

Smoothing a braid isotopy with all seams fixed. First choose smooth representatives at the two ends by [F1] and give them endpoint collars by the fixed height reparametrization of [F3], and concatenate their approximation homotopies with the given family. The empty family needs no approximation. For $n\ge 1$ the disk-boundary distance has positive uniform minimum on the compact parameter square; for $n\ge 2$ include the finitely many pairwise strand distances as well. Use the boundary minimum alone at $n=1$. Reparametrize height and family parameters to make the family constant in height collars and equal to the two smooth end braids in family-parameter collars. Approximate its finitely many real coordinate functions by tensor-product Bernstein polynomials on the square. For a continuous scalar function $f$ on $[0,1]$ and $N\ge 2$, with out-of-range binomial coefficients taken as zero, the binomial weights sum to one, have mean $t$ and variance $t(1-t)\mathrm{/}N\le 1\mathrm{/}(4N)$, by the identities $k\left(\genfrac{}{}{0px}{}{N}{k}\right)=N\left(\genfrac{}{}{0px}{}{N-1}{k-1}\right)$ and $k(k-1)\left(\genfrac{}{}{0px}{}{N}{k}\right)=N(N-1)\left(\genfrac{}{}{0px}{}{N-2}{k-2}\right)$. Uniform continuity gives error at most $ϵ$ on $\mathrm{\mid }k\mathrm{/}N-t\mathrm{\mid }<\delta$. The remaining weight is at most $1\mathrm{/}(4N{\delta }^2)$, since its squared deviation is at least ${\delta }^2$ per unit weight. Thus the total error is at most $ϵ+2\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}(4N{\delta }^2)$, uniformly in $t$, and tends to zero. Apply this to each of the finitely many real coordinates, successively in the two variables; convex averaging is a contraction for the uniform norm, so the two errors add. This proves the required uniform square approximation. Choose error smaller than one tenth of that separation. Repair each family-parameter edge by adding a smooth collar cutoff times the difference between the prescribed smooth edge and the approximant's restriction to that edge; the two family collars are disjoint, and these corrections have norm at most the approximation error. Then repair the two height edges to their fixed points by the analogous disjoint height cutoffs. On the family edges these latter corrections vanish because the prescribed end braids already have the correct height endpoints. The resulting map is smooth on the square, fixes all four edges, and differs from the collared continuous family by less than five times the chosen error, so remains in the disk with all strands distinct. Finally compose its height variable with a smooth map constant near $0,1$ and equal to the identity outside the original fixed height collars, and its family variable with one constant near $0,1$; this makes all height jets agree with the fixed endpoints and all family end collars constant, without changing the braids at those ends. The same uniform margin permits straight interpolation to the collared family. Thus this is a smooth braid isotopy between the chosen smooth representatives, with the one fixed endpoint permutation throughout.

Arbitrary chosen models and independence. Choose any smooth endpoint-flat models ${\beta }_s,{\beta }_s^{\mathrm{\prime }}$ of the given braids. Their endpoint-fixed approximation isotopies, the given $Z$, and the inverse approximation isotopy form a continuous braid family between them. Step 1.2 makes this into a smooth family of collared models, and step 1.1 gives ambient equivalence of their selected literal smooth closures. If either chosen model is smooth with matching jets but lacks constant collars, the explicit height interpolation of the Definition [F3] preserves all matching cycle-seam jets, and the compact-cycle argument of step 1.1 supplies the same equivalence. Thus the result holds for every permitted choice of model. Applying the same argument with ${\beta }^{\mathrm{\prime }}=\beta$ proves independence of the smooth-category closure class in [F3]; it is established here, rather than assumed from the Definition. No ambient smooth isotopy of a nonsmooth raw image is used. All constructed model images remain closed $n$-braids about $A$.

Conclusion. Every braid isotopy from $\beta$ to ${\beta }^{\mathrm{\prime }}$ therefore yields an ambient isotopy of $S^3$ carrying $\widehat{\beta }$ to $\widehat{{\beta }^{\mathrm{\prime }}}$, so the closure construction factors through the braid isotopy class; the two uses of ${\mathrm{A}\mathrm{C}}_{\omega }$ are exactly the smoothing of [F1] and the ambient isotopy extension of [F2].