Non-braid-like Reidemeister moves are generated by braid-like moves and reductions

The non-braid-like R2 on one circle. Read the five panels in the top row of Traczyk's Figure 3, printed p. 411. The upper arc points left and the lower arc right. Name their outside ports $U_L,U_R,L_L,L_R$; the outgoing ports are $U_L,L_R$. Since both smoothed arcs belong to one circle $C$, the untouched outside paths join $U_L$ to $L_L$ and $L_R$ to $U_R$. The first arrow adds one R1 curl to the lower arc, producing a small circle $C_1$ coherent with $C$. The second arrow adds a second, oppositely signed R1 curl on that curl. Oriented smoothing now gives the old continuation, the middle diamond circle $C_1$ and the upper loop circle $C_2$. Work on the side of $C$ containing the move disk, choosing the pole on its other side only to read the annulus orientations: $C_1,C_2$ are side by side there, $C_1$ has the boundary direction of $C$, and $C_2$ the opposite direction. Thus $C_1$ is coherent with $C$, $C_2$ with $C_1$, and $C_2$ is incoherent with $C$. The third arrow pulls the upper loop through the upper horizontal arc, which is over both new crossings in the displayed picture. This antiparallel R2 is a reducing move on $C,C_2$, not on the first coherent pair: smoothing gives its empty small circle $C_a$ above the horizontal arc and the other new circle $C_z$ surrounding the unchanged $C_1$. The last arrow cancels the two lower self-crossings by R2. Its inverse is a reducing move: smoothing after cancellation gives two side-by-side circles of the same orientation, hence an incoherent pair, while smoothing before cancellation gives the coherent pair $C_1,C_z$; $C_a$ is unchanged. The two reducing strips are different. The crossing counts are $0,1,2,4,2$, and the last panel retains exactly the two desired crossings between the original arcs and all four outside germs. The sequence is therefore R1, R1, reducing R2, inverse reducing R2. The two R1 signs are opposite, as are the crossings in each R2 pair. Reflecting the spatial height makes the upper arc under both target crossings; reversing the two arc orientations and running the sequence backwards supplies the remaining sign, orientation and inverse variants. The outside paths and all other Seifert circles stay fixed throughout.

The antiparallel R2 on distinct circles. Before creation of its two crossings the opposite local directions on the exposed boundaries make the two circles incoherent in their common annulus. The move disk consequently supplies a reducing arc, and its R2 is precisely a reducing move, with either over/under choice. Running this creation backwards gives the inverse reducing move. A coherent pair is not made incoherent merely by reversing crossing signs.

The non-braid-like R3. Choose the arc with orientation opposite to the other two as the detouring arc. In the bottom square of Figure 3, pull a short part of that arc past one branch by an antiparallel R2, perform the now braid-like R3 at the other end of the detour, and remove the detour by the reverse antiparallel R2. Tracking the three original pairwise crossings leaves exactly the target R3 picture. If a detour R2 acts on one Seifert circle, replace it by step 1.1; if it acts on distinct circles use step 1.2. The consistent total height orders choose which detour is over or under. Choosing the odd-oriented arc in each orientation pattern, and reversing or reflecting the pictured square as needed, covers all non-braid-like variants.

Conclusion. Steps 1.1, 1.2 and 2.1 show that every non-braid-like R3 and every non-braid-like R2 is generated by the four allowed move types, while type I moves are separately allowed and braid-like II and III moves are already in the move list. Since by Reidemeister's theorem for oriented diagrams any two diagrams of the same oriented link are connected by the oriented moves R1, R2, R3, replacing each non-braid-like R2 and R3 by the compositions above and keeping the remaining moves gives the required sequence through the four move types. AC is inherited from [F1] and supplies the countable choice used by Reidemeister equivalence (AC implies DC implies countable choice).