Defect regions, reducing arcs and the Yamada-Vogel reducing move

Assume the Axiom of Choice. Let $D$ be an oriented diagram with Seifert picture $S$, consisting of the Seifert circles $C_1,\dots ,C_m$ and the finite set of signed arcs recording the crossings (Seifert smoothing and Seifert circles of an oriented link diagram), and let coherence of pairs of Seifert circles be as in Coherence of Seifert circles and the height of a diagram.

A region of $S$ is a connected component of $S^2\setminus S$, where $S$ includes both the circles and the signed arcs. Thus a region contains no signed arc in its interior. Its boundary can contain signed arcs and portions of Seifert circles. A region is a defect region if two Seifert circles $C_i\mathrm{\ne }{C}_j$ that are incoherent both occur in the boundary of that region. The existence of a defect region when the height is positive is the content of the defect-region lemma below.

A reducing arc is a simple arc $\alpha$ contained in a defect region together with its endpoints, joining an incoherent pair $C_i,C_j$ of Seifert circles and meeting the union of the Seifert circles exactly in its two endpoints; the arc may be taken polygonal inside the region.

The Yamada-Vogel reducing move performed along $\alpha$ slides one of the two circles, say $C_i$, over the other along $\alpha$: in the diagram this is a Reidemeister II move of the original diagram in which a neighbourhood of the arc is replaced by the standard band picture of two crossings of opposite signs, so that in the new Seifert picture the incoherent pair $C_i,C_j$ is replaced by two coherent Seifert circles $C_a,C_z$ joined by two signed arcs of opposite signs, all other Seifert circles unchanged, $C_a$ bounds a disk containing no other new Seifert circle, and $C_z$ bounds a disk containing all Seifert circles that were contained in the annulus cobounded by $C_i$ and $C_j$. The inverse of a reducing move is also allowed. A diagram is reducible when it admits a reducing arc; the move is defined for every defect region and the resulting picture is again a Seifert picture of an oriented diagram of the same link.

The choice axiom is used exactly where coherence is used, through the annulus lemma of Two disjoint circles in the two-sphere cobound an annulus; the local band picture of the move itself is an explicit Reidemeister II replacement of two oppositely signed crossings.