A height-zero diagram represents a closed braid

The complementary dual tree. Make one vertex for each component of the complement of the $n$ circles in $S^2$ and one edge for each circle, joining its two adjacent components. With no circles this is one vertex. Inserting a new disjoint Jordan circle splits one existing region into two, by [F3], and replaces that vertex by two vertices joined by one edge. Induction therefore gives a tree with $n$ edges. Suppose a vertex has three incident circles. Orient its common region and let ${ϵ}_i\in \{+1,-1\}$ record each given circle orientation relative to the induced boundary orientation on that region. For any pair of those boundary circles the annulus between them includes the common region, so its boundary orientation there is the same; coherence [F1] requires ${ϵ}_i=-{ϵ}_j$. Three such signs cannot all be pairwise opposite. Thus every vertex has degree at most two and the tree is a path. Choose a pole in either end disk and use stereographic projection from that pole. The circles are now nested in the remaining plane. Their annulus orientations show that all are oriented in the same angular sense. This argument never assumes the original diagram or Seifert graph is connected.

Crossing strips occur only between successive circles. At an oriented smoothing the two arcs bounding its strip are parallel and point in the same direction by [F2]. If they belonged to the same Jordan circle, the connected strip interior would lie on the same complementary side of that circle at both arcs. But equally directed arcs on opposite sides of this strip have that interior on opposite local sides; the inside of an oriented Jordan circle lies consistently on one local side. This contradicts separation [F3]. Hence a strip joins distinct circles. Its connected interior lies in one complementary component, so in the path of step 1.1 it joins precisely two consecutive circles across their common annulus. Within one such annulus, disjoint strips have the same cyclic order of their endpoints on the two coherently oriented boundaries: cutting along one strip makes a disk, and two further strips with interlaced endpoints would cross by Jordan separation.

Smooth straightening, rather than merely a topological change of coordinates. We give the local construction needed to straighten the finite picture. A smooth simple curve has a tubular collar [F4]. A sufficiently fine inscribed polygon, with its corners rounded in pairwise disjoint small disks, is a smooth normal graph over that curve: on a compact regular arc its tangent varies uniformly little on a sufficiently fine partition, so normal projection is locally strictly monotone; the collar and compact separation of nonadjacent arcs make it globally one-to-one. Multiplying its smooth normal displacement by a parameter is an isotopy through smooth normal graphs, extended by [F4]. A polygonal disk can be triangulated by induction: at a convex vertex, if its adjacent-vertex triangle contains no other vertex use that diagonal; otherwise choose a vertex in that triangle farthest towards the convex corner from its opposite edge and use the resulting interior diagonal. A vertex or edge blocking this latter diagonal would contradict the maximal choice; separation keeps its interior in the disk. Each diagonal splits the polygon into smaller polygons, so the induction terminates. To straighten a curve around a fixed interior point, root this triangulation at the triangle containing that point, after a sufficiently small generic perturbation of the polygon vertices so no vertex-pair line passes through it. Such perturbations avoid finitely many proper line conditions, preserve simplicity and keep the point inside by compact separation. Remove leaf triangles in reverse order, leaving the root triangle. Each removal replaces a two-edge boundary arc by its diagonal in the empty triangle. In affine coordinates the two-edge ear is a graph. Choose its rounded version and the rounded diagonal with identical smooth endpoint germs; interpolate their graph functions smoothly, retaining these germs. Every parameter slice is therefore smooth, including the initial and final slices; the literal polygon is only the combinatorial guide. To realize a small graph displacement $u(s)$ ambiently, use the smooth collar map $(s,r)\mapsto (s,r+\chi (r)u(s))$ with $\mathrm{\parallel }{\chi }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}<1$. Split a larger displacement into finitely many small ones inside the clear local graph neighbourhood. These are explicit smooth diffeomorphisms, fixing the collar boundary and the unchanged germs. At every time it is an embedded smooth arc, and no other portion of the curve meets that neighbourhood. The fixed interior point is untouched. The remaining rounded triangle is star-shaped about that point and is taken to a small concentric circle by positive radial interpolation, again with support away from the point. By [F4] these finitely many smooth curve isotopies are ambient isotopies. Apply them first to the outermost nested circle, transporting its entire interior picture, and then successively inside the disk bounded by each already fixed round circle. Choose the centre $p$ inside the innermost disk; the construction fixes $p$ throughout. This puts all circles concentrically about $p$. The same triangle-arc operation straightens finitely many disjoint crossing strips in an annulus: cut along a spanning strip, leave small neighbourhoods of the strip endpoints and their smooth incident circle/arc germs fixed. Approximate only the remaining compact smooth arc pieces in their disjoint normal collars by rounded polygonal graphs, as above. Each remaining proper strip core cuts the disk into two disks by Jordan separation and Schoenflies. Cutting successively along their disjoint cores gives finitely many disk faces, each triangulated by the preceding diagonal construction; remove the resulting triangles between each core and its target, fixing endpoint collars and already fixed cores. Once a core is straight its thin strip is straightened in its normal collar. All supports miss the other fixed cores. The relative disk-face boundary pieces retain their original smooth germs, and the rounded graph interpolation and explicit collar diffeomorphisms apply to the moving interiors; neither a polygonal endpoint nor a cornered whole face is used as the source of smooth isotopy extension. Round corners and use fixed collars as above. An integral winding of its first spanning strip is removed before this construction by $(r,\theta )\mapsto (r,\theta -2\pi kf(r))$, where $f$ is smooth, equals $1$ at the inner boundary and $0$ at the outer boundary. Its isotopy uses parameter multiples of $f$ and rotates the entire inner disk at the same time, so is a sphere isotopy; it does not require a boundary-fixed annulus isotopy of a nonzero Dehn twist. Thus every straightening used here is realized through smooth embeddings and [F4], not inferred from the homeomorphism assertion of Schoenflies.

Choose compatible crossing-event angles. Starting with the outermost circle, choose distinct angular positions for its crossing-strip endpoints in their cyclic order. Pass inward through the circle chain. The endpoints shared with the preceding annulus already have angles, and by step 2.1 their order agrees on both boundaries. Insert the new events for the next annulus into the appropriate open gaps in that finite circular order, avoiding all previously chosen event angles. If there are no shared events, choose any initial angles in the specified cyclic order. An orientation-preserving smooth reparametrization of each circle realizes these finite assignments: on the intervening open arcs choose positive smooth derivatives with the prescribed integrals, using matching endpoint collars. The annulus strip construction in step 3.1 then takes each crossing strip to a small radial strip at its assigned angle, while the remaining annulus pieces match the two boundary reparametrizations. To interpolate prescribed boundary circle coordinates across a strip-free annulus, take increasing lifts $f_0,f_1$ with $f_i(\theta +2\pi )=f_i(\theta )+2\pi$ and use the angular coordinate $(1-\chi (r))f_0(\theta )+\chi (r)f_1(\theta )$, where $\chi$ is smooth and constant near both ends; its angular derivative is positive. The strip-relative construction gives the same boundary collars, so the finitely many constructions glue smoothly. No connectedness assumption is needed; an annulus with no strips can use any interpolation between its boundary coordinates.

Recover the closed braid and read its word. In each radial crossing strip put back the original signed crossing, using the model of two strands with increasing angular parameter and one signed half twist in the radial-depth disk. Outside the strips use the concentric oriented circles. Step 1.1 makes their angular directions agree, and a chart of the opposite orientation, if necessary, chooses that direction as increasing height. When reversing the planar orientation, reverse the depth coordinate as well. The combined three-dimensional coordinate change preserves ambient orientation and represents the same oriented link; read its over/under information in these new depth coordinates. The finitely many strips have distinct event angles by step 4.1. Cut at an unused angle and read their half twists in angular order; reverse the chronological list when using the rightmost-first geometric product convention. This is an explicit finite Artin word in $B_n$. Its geometric representative [F5] and the recovered diagram have exactly the same oriented strands and signed strip models. To transfer the sphere isotopies to link equivalence, place a lift of the diagram in a thin normal collar of the projection sphere, keeping the higher branch higher in each crossing strip. Compressing each normal fibre by a positive factor preserves these strict orders and is an isotopy of the link. A smooth sphere isotopy lifts in that collar by its tangential velocity field, extended constantly along sufficiently short normal fibres and cut off farther away; equivalently apply [F4] to the projection sphere and choose this collar extension. It transports every crossing strip without interchanging its depths. Its ambient extension therefore carries the original oriented link to this closed braid. The remaining local strip lifts with the same projected arcs and strict depth orders are joined by linear interpolation of the depths, which preserves embeddings. Thus the standard closures agree up to ambient isotopy. Each open strand period advances once around the pages; a closed component may advance several periods according to its permutation cycle, as [F5] specifies. There is no assertion that every component has degree one.

Boundary cases and conclusion. If $n=0$, the diagram and strip set are empty and [F5] gives the empty word and empty closure. If $n=1$, step 2.1 rules out every strip, and step 3.1 takes its sole circle to a round circle; the empty word in $B_1$ closes to that unknot. For every $n>1$, steps 1.1-5.1 give the sphere isotopy, the chosen planar chart and the explicit word with the original oriented link as closure. All constructions are finite; AC is inherited only from [F1], [F3], [F4]. The preliminary change of chart is allowed on $S^2$, and is not asserted to be a planar isotopy of the original fixed chart.