Marked concordances admit finite movies with a protected mark

A monotone target for the marked arc. Choose a spatial stereographic chart containing the compact spatial image of $\gamma$: a smooth arc has null image in the three-sphere by the same Sard dimension argument used in [F3]. Write $\gamma (u)=(b(u),h(u))$ there and choose its collar parametrization so $h(u)=u$ near zero and one. The arc ${\gamma }_{\ast }(u)=(b(u),u)$ is embedded because its time coordinate is $u$. The interpolation $G(s,u)=(b(u),(1-s)h(u)+su)$ is a smooth homotopy fixed near the two endpoint collars. Away from those collars its time coordinate stays in $(0,1)$. It need not initially be an isotopy, so an additional general-position argument is necessary.

The codimension-three arc isotopy is actual. Choose smaller fixed $u$ collars and leave small $s$ endpoint collars unperturbed. Those $s$ collars are already embeddings, because the initial and final arcs are embeddings and that is an open $C^1$ condition on a compact interval. Work in the global ${\mathbb{R}}^3\times \mathbb{R}$ chart of step 1.1. Let $\beta (s)$ be zero on smaller parameter-end collars and positive throughout their active complement; let $\chi (u)$ be zero on smaller source-end collars and positive throughout their active complement. The explicit eight-real-parameter family $G_{A,B}(s,u)=G(s,u)+\beta (s)\chi (u)(A+uB),A,B\in {\mathbb{R}}^4,$ fixes every protected collar. At one movable $u$, its derivative parameter coefficients are $\beta {\chi }^{\mathrm{\prime }}$ and $\beta (\chi +u{\chi }^{\mathrm{\prime }})$, not both zero, so it prescribes ${\mathrm{\partial }}_{u}G$ freely. At two distinct movable values $u,v$, the value-coefficient matrix has determinant ${\beta }^2\chi (u)\chi (v)(v-u)\mathrm{\ne }0$, so their values are independent. If one point is in a fixed collar, the other point alone prescribes their difference freely; two fixed collar pieces cannot collide. The zero derivative stratum has codimension four on the source $(s,u)$ of dimension two. The collision stratum $G_{A,B}(s,u)=G_{A,B}(s,v)$ has codimension four on the open configuration source $(s,u,v)$, $u\mathrm{\ne }v$, of dimension three. On the portions where parameters are zero, the already embedded compact collar families exclude these defects after a sufficiently small perturbation. On all other portions the just-verified evaluations are submersions. By [F1] a common arbitrarily small parameter avoids both strata. Thus every slice is an immersion with no coincident distinct pair. There is no hidden near-diagonal collision: compactness and the immersion lower bound give a uniform local injectivity neighborhood; the pair argument already treats its complement. Each compact slice is an embedding. Choose parameters small enough to keep the active track inside the ambient interior; this follows from its positive distance to the time boundary off the source collars. The result is an actual embedding isotopy from $\gamma$ to ${\gamma }_{\ast }$. Extend it by [F2], choosing the extended velocity zero near smaller end collars, and apply it to the whole $S$. It makes the marked arc monotone while retaining the collared marked boundary.

A product strip about the track. On the new surface, height has derivative one along ${\gamma }_{\ast }$. The implicit-function theorem supplies a neighborhood with coordinates $(t,v)$ in which height is exactly $t$ and the marked arc is $v=0$. Compactness gives one uniformly narrow strip; in the end collars it agrees with the specified product strip. In each spatial level its strip is an embedded interval through $b(t)$. Its nonzero spatial tangent at $v=0$ has a smooth unit direction $a(t)$. Complete that direction to a smooth spatial frame along the interval: start with one frame and transport an orthogonal frame by the smooth projection-commutator ODE, or solve the equivalent finite-dimensional orthogonal transport ODE. The interval has no monodromy obstruction. In its fiberwise tubular coordinates the strip differs from $v\mapsto b(t)+va(t)$ by $O(v^2)$ with its first derivative $O(v)$, uniformly in $t$. Interpolating this error to zero inside a smaller strip, with a spatial cutoff in a slightly larger moving ball, is an embedding isotopy for a sufficiently small uniform width. Its spatial vector field extends and integrates as in [F2], with time $t$ merely a smooth parameter. Hence a level-preserving spatial coordinate family sends the marked track and a surrounding surface strip to a fixed straight model in one small three-ball. Choose the endpoint coordinate identifications to be its endpoint values; compatible prescribed long presentations need no change there. These are coordinate identifications, not an assertion that distinct physical endpoint marks were already the same point.

Puncture at the mark and retain the collars. Choose the fixed model so its central point is the stereographic point at infinity and its surface interval is a round circle through that point. Stereographic projection makes the punctured interval two fixed straight exterior rays. Remove a smaller open marked strip. The rest of the surface is a compact collared tangle cobordism in a fixed ball of ${\mathbb{R}}^3\times [0,1]$, with two fixed endpoint collars on the spatial ball boundary at every time. All other parts of the surface are disjoint from the retained strip; compactness after taking a slightly smaller strip provides a uniform separation. Extend the two spatial end rays by one fixed planar exterior closing arc outside a sufficiently large projection disk, marking a point on that arc. This closure represents exactly the deleted standard marked interval in its three-ball, with the same endpoint parametrizations. The compact part projects inside the large disk and the closing arc misses its interior. Consequently all later compactly supported projection events can occur inside that disk while the mark and a neighborhood of it remain fixed outside.

Finite Morse events relative to the strip. Run the height part of the construction in [F3] on the compact tangle part. All bumps have support outside the fixed tangle-end collars and the removed marked strip. Where those collars are protected, height is the product coordinate and has no critical point. On the remaining finite chart cores the bump-linear perturbations of height span each cotangent fiber exactly as in [F3]; [F1] therefore gives a Morse height. The critical points are finite, their values can be separated by disjoint supported bumps, and sufficiently small perturbations remain embeddings with an actual ambient trace isotopy. Their tangent-plane projections and complementary regular branches can be made generic by spatial bumps on the same compact part. The resulting birth, saddle and death disks lie inside the large projection disk. The long end collars and the marked closing arc are fixed. These are the actual local models of [F3], with a protected regular boundary added, not any assumption that a mark survives an arbitrary unbased movie.

Finite projection events and the actual trace. On each complementary regular-height slab choose the normalized gradient equal to the fixed product field near the long ends; patching the product field with the normalized field preserves $dh(X)=1$. Its flow gives an isotopy of compact tangles fixed near their ends. Flatten small strips at the cut times by the interpolation $(F_{(1-s)t+s\rho (t)}(x),t)$ from [F3]; its time coordinate is still $t$, so it is an embedding isotopy. Apply the finite-jet proof in [F4] relatively to the fixed exterior arc and end collars. The needed parameter bumps prescribe independent jets at the moving source points. No forbidden event can involve a point on the exterior arc and a compact point, because their projections are separated by the large disk; the fixed joining collars are straight, with a protected regular neighborhood. Events wholly in the compact part have exactly the evaluation submersions and deleted-diagonal bounds of [F4]. Hence there are finitely many ordinary R1/R2/R3 events, fixed near slab ends; identify them by the second supplier in [F4]. Small spatial interpolation retains the actual traced surface, and [F2] extends it relative to every protected collar. Thus all the resulting Reidemeister support disks also miss the marked arc.

Glue the finitely many critical intervals and regular slabs along their stationary cuts. Choose their exceptional times distinct as in [F3]. The fixed exterior closing arc supplies the mark in every frame, including any frame with additional circles, and none of its points participates in a birth, saddle or death. Undo the spatial coordinate family of step 3.1 and the marked-strip replacement to compare the trace: all actual perturbations were ambient isotopies relative to the retained collars, and the coordinate family has the explicitly recorded endpoint identifications. Therefore the transported trace is the original marked concordance up to the asserted relative isotopy. The construction preserves the annulus and its orientation. It proves existence for freely chosen compatible based endpoint presentations; no passage between arbitrary prescribed planar diagrams or arbitrary basepoint presentations, and no equality of algebraic movie maps, has been used.