Braid-like moves can be moved to height zero

A reduction away from a local move disk. For a type I move adding a kink, choose a disk $U$ meeting the initial Seifert picture in one proper arc. For a braid-like II or III move, smoothing inside its standard move disk gives parallel, coherently oriented proper arcs; signed crossing arcs lie between adjacent arcs. Each inside component of a complementary region is consequently a cap or a terminated strip meeting one outside component, or an uncut strip between two coherent circles $E,F$ which can join two outside components. Take a defect region $R$ supplied by [F5]. If a component of $R\setminus U$ exposes an incoherent pair, an arc inside that component is the required reduction away from $U$. Suppose none does. Every outside component then exposes at most two distinct circles, and if there are two they are coherent: indeed three exposed circles have two with the same orientation as boundary components of the region, by the two possible boundary orientations, and those two are incoherent in their cobounding annulus. Any outside component attached to an uncut strip exposes its pair $E,F$ and therefore cannot expose a further circle. Thus along every chain of uncut strips all outside components expose the same coherent pair. Caps and terminated strips cannot join different outside components or introduce a further circle; components wholly inside $U$ also expose at most a coherent pair. Since $R$ is connected, it would expose at most this coherent pair (or a single circle if there is no joining strip), contradicting that it is a defect region. A reduction away from $U$ therefore exists. This establishes avoidance only for the specified local move disks, with deletion of a kink handled by reversal.

Pushing the move to height zero. Let a type I move adding a kink, a braid-like II move or a braid-like III move $t$ start at positive-height $Y$. By step 1.1 choose a reducing move $r$ disjoint from its local disk. The supports are disjoint, so $r$ and $t$ commute. Replace $Y\stackrel{t}{\to }tY$ by $Y\stackrel{r}{\to }rY\stackrel{t}{\to }trY\stackrel{r^{-1}}{\to }tY$. The copy of $t$ starts at height one lower by [F3]. Repeat at each positive starting height; after finitely many reductions this copy of $t$ starts at height zero, surrounded by the reductions and their inverses. For a type I deletion, reverse the corresponding adding construction.

Height-zero moves and the type I return path. Height zero alone does not make a diagram a closed braid in its original planar chart. First normalize its Seifert picture by [F6], transporting the local move disk as well. For II and III the two smoothed local pictures agree up to isotopy of their coherently directed proper arcs. In the straightening construction of [F6], retain these arcs as marked pieces and put their crossing strips in one small angular interval; outside that interval the two pictures use the same nested circles and strips. The local parallel arcs can be straightened in a rectangle with fixed boundary germs, so both recovered diagrams are closed braids and the transported move is braid-like. Now [F2] applies, giving a braid isotopy, hence a conjugation by [F4]. For type I retain the marked arc and the side on which its kink is to be added. In the circle chain, choose the end disk lying on that side of its supporting circle, and normalize with the axis in that end disk; either end disk is allowed by [F6]. Thus an outward kink on the innermost circle in the old chart uses the opposite end disk, beyond all the enclosing circles. Let $k$ be the number of circles between the marked arc and this chosen end disk. Carry a short part of the marked strand beneath those $k$ strands by braid-like II moves, add the kink on the end-disk side of the now nearest strand, and return it under the same strands. The end-disk-side kink smooths to a new circle in that end disk, coherent with every old circle, so both diagrams at this stage have height zero; with the end strand indexed last and a cut at the kink, its reading is the signed Markov stabilization of [F4], up to cyclic conjugation. Crossing sign determines the sign of stabilization independently of the selected side. The return across each strand consists of a braid-like III move followed by an inverse reducing move, as in the survey's printed p. 20 prescription and the last two arrows of Traczyk's Figure 4: the new kink circle and the crossed old circle are distinct, and the final antiparallel II cancels the temporary transport crossings. The marked outside germs and the prescribed kink side are restored. For $k=0$ this is a stabilization directly; for $k>0$ the inverse reductions can create positive height, even when the original marked strand was nearest the old axis. Nearest-strand position alone therefore does not suffice to identify a type I move with stabilization. Apply step 2.1 to each braid-like II or III move introduced in this finite return path, and reverse the construction for an inverse type I move.

The replacement sequence. Replace every non-braid-like II or III move using [F1], then replace the resulting type I and braid-like II and III moves by steps 2.1 and 3.1. The braid isotopies and Markov moves now occur in the normalized height-zero pictures. Undo each normalization before resuming the original reducing path. Sphere isotopy preserves coherence and transports reducing arcs, so these insertions do not change any height or reducing endpoint. Cut each remaining finite sequence of reducing moves and inverses at every height-zero diagram. Every resulting portion has positive-height intermediate diagrams; its endpoints can have height zero, including the last reduction from height one to zero and the first inverse reduction from zero to one. This gives the stated grouping of reducing content with the height-zero moves.

Conclusion. Steps 1.1-4.1 exhibit the required sequence and show that all braid-like content may be assumed at height zero. AC is inherited from the coherence and reducing-move apparatus of the height function.