Reidemeister moves between closed braid diagrams factor through Markov moves

Reidemeister sequence and phase one. By the oriented Reidemeister equivalence theorem there is a finite sequence of planar isotopies and oriented R1, R2, R3 moves from $B$ to $C$; by [F5] this uses ${\mathrm{A}\mathrm{C}}_{\omega }$, discharged from AC. Apply [F1] to every non-braid-like R2 and R3 in the sequence: the sequence is replaced by one using only type I moves, braid-like R2 and R3 moves, and reducing moves and their inverses.

Phase two: braid-like content at height zero. Apply [F2] to replace each type I and braid-like II or III move. Use the sphere/chart normalizations in [F2] before applying [F3]; height zero in the original chart alone is insufficient. Keep the coherent oriented circle order and a transported cut in each normalized picture. The finite crossing-order argument for planar readings in [F3] compares compatible straightenings by far commutations and cyclic conjugations; if the two end disks are exchanged, strand-index reversal is the old-strand conjugation of [F7]. Thus normalization choices change only conjugacy of the readings. Take the endpoint normalizations to be the given closed-braid forms of $B,C$. The II and III moves now occur on closed braid diagrams and are braid isotopies by [F3], hence conjugations. The type I replacement has a stabilization or destabilization at height zero and, when the kink must return across other strands, includes braid-like III moves and inverse reductions as specified by [F2]. Group the resulting transformation into height-zero Markov portions interleaved with reducing sequences. Split each reducing sequence at its height-zero diagrams, so its intermediate diagrams have strictly positive height and its endpoints have height zero.

Phase three: track a fixed maximum height. Consider a reducing portion from step 2.1, with maximum height $H$. If it is empty there is nothing to eliminate. Otherwise $H>0$ and each occurrence of $H$ is an interior peak by [F6]. Track the constructions of [F4], rather than assuming a height bound in its alternative conclusion. At a peak $X\leftarrow Y\to Z$, inserting a reduction $t$ replaces it by $X\leftarrow Y\to Y(t)\leftarrow Y\to Z$; $Y$ still has height $H$ and every new neighbour has height $H-1$. The arc surgeries in [F4] first decrease intersection numbers of each resulting pair until they are at most one, then replace one-intersection pairs by disjoint pairs. These are finite refinements at this same $Y$, never insertions above $H$. A compatible disjoint pair is replaced by its common double reduction of height $H-2$. An available third compatible arc gives two such valleys. For $H=1$, the height-one exchange construction of [F4] removes the peak at height zero. The remaining disjoint pairs at $H\ge 2$ are irreducible four-band peaks. For each use [F7]: descend from both neighbours, which start at $H-1$, to height zero, compare there by its finite Markov sequence, and reverse the second descent. By [F6] that entire replacement has height at most $H-1$. After the finitely many refinements and replacements at all original height-$H$ occurrences, no height-$H$ diagram remains. Split at the inserted height-zero Markov portions. Every remaining reducing portion has smaller maximum, so induction on the nonnegative integer $H$ eliminates it. This yields a finite Markov sequence between the normalized endpoints.

Conclusion. Steps 1.1-3.1 factor the chosen Reidemeister sequence between $B$ and $C$ through Markov moves, so the braids read from $B$ and $C$ are Markov equivalent. AC is inherited through the AC-stated generating, height and four-band items.