Seifert smoothing and Seifert circles of an oriented link diagram

Let $D$ be an oriented link diagram (Regular oriented link diagrams), regarded as a decorated immersed oriented $4$-valent planar graph, and let $V$ be its finite set of double points. At each $v\in V$ the two branches of the diagram cross; the local Seifert smoothing of $D$ at $v$ replaces the crossing by the unique pair of non-crossing arcs in a small disk about $v$ that joins the four half-edges in the orientation-respecting way: entering along a strand, one follows the smoothing turn that keeps the travelling direction consistently on the same side, so that the two oriented arcs through $v$ are reconnected without crossing. Performing this replacement in pairwise disjoint small disks, one for each crossing, and leaving the rest of the diagram unchanged produces the Seifert smoothing of $D$: a finite disjoint union of oriented simple closed curves in $S^2$, its Seifert circles. At each crossing the smoothing reconnects the four half-edges in one of the two possible non-crossing ways, and exactly one of the two respects the orientations; hence the smoothing is a well-defined operation on the diagram. The Seifert smoothing may merge two circles into one or split one circle into two at a crossing, so its number of circles is a new invariant of the picture and is not in general the number of components of the link.

The Seifert picture of $D$ is the smoothed picture together with one signed short arc at each former crossing $v$, drawn transversely to the two smoothed strands and joining the two distinct Seifert circles that passed through $v$; the sign of the arc is the sign of the crossing, positive or negative. The signed arcs lie in the complement of the Seifert circles and meet the circles only at their endpoints (Embedded submanifolds and slice charts); every crossing is recorded exactly once. The two smoothed arcs at a crossing are parallel and equally directed. They cannot belong to the same oriented Jordan circle: the connecting strip is on opposite local sides of its two equally directed boundary arcs, whereas a fixed complementary side of an oriented Jordan circle is consistently on one local side. Thus each signed arc joins distinct circles, coherently oriented in their common annulus. The orientation of each Seifert circle is the one induced by the Smoothed strands, and the collection of circles and signed arcs is the object to which the coherence and height definitions (Coherence of Seifert circles and the height of a diagram) are applied.