Tait flyping theorem

Identify the two represented oriented links with one smooth link $L$ by the given isotopy. It is nonsplit by [F1]. Full AC supplies the choice hypotheses of [F2] and [F5]. The identification retains the actual oriented component correspondence to be used in all capped factors.

In either diagram choose a smooth simple circle meeting the diagram transversely in two ordinary points whenever both sides contain crossings. By [F7] its two sides are smooth disks, so cap each cut arc by a crossing-free arc in its cap disk, carrying the endpoint pairing and orientation. Both capped shadows are connected: an additional shadow component on either side would already have been a separate component of the original shadow. Their crossing sets partition that of the piece being cut. The capped diagrams are alternating. The one proper projected arc has an even number of crossing encounters: its self-crossings contribute two visits each, and its intersections with each closed projected component are even. For the latter, decompose the closed component's finite mod-two edge set into simple cycles. Each cycle separates by [F7], and the proper arc begins and ends on the disk boundary outside those cycles, so it crosses each cycle evenly. The first and last encounters on the proper arc therefore have opposite over-under status. Closing its ends introduces no new encounter.

If $L$ is empty, both diagrams are empty, and the identity sphere planar isotopy proves the conclusion. Henceforth assume $L$ is nonempty. If $L$ is the unknot, [F2] forces both diagrams to have zero crossings. A connected crossing-free diagram is one embedded oriented circle on the projection sphere, and the two such circles are carried to one another by a sphere planar isotopy with their orientations transported. This proves the conclusion in this case. Henceforth $L$ is nontrivial and both diagrams have positive crossing count.

The capped diagrams are reduced. Indeed, a nugatory separator through an old crossing in a capped diagram can be moved off the crossing-free cap disk: each part in that disk lies in one of the two complementary disks of its cap arc and can be pushed to the boundary collar. It would then give a nugatory separator through the same crossing in the original piece, contrary to reducedness. The cut and caps have smooth product collars. In the represented link they are the usual twice-punctured decomposition sphere and its standard trivial-arc cap balls.

Repeat these cuts in the capped pieces, retaining their cap-ball parameters and attachment collars. A later cut is moved off the interiors of earlier cap balls using their crossing-free trivial-arc product collars; equivalently shorten each artificial cap into its boundary collar before choosing the next cut. Lifting the resulting finite diagrammatic connected-sum decomposition gives pairwise disjoint decomposition spheres in the original link. Termination is finite: if a piece with $a+b$ crossings is replaced by two pieces with $a,b>0$ crossings, the sum of squared crossing counts falls by $2ab$. The final positive-crossing capped diagrams are therefore connected, reduced, alternating and have no two-point splitting circle with crossings on both sides. Crossing-free trivial connectors are suppressed while retaining all other attachment collars.

Every positive-crossing terminal factor is nontrivial by [F2]; otherwise its reduced alternating crossing count would be zero. It is nonsplit by [F1] and prime by [F4]. Thus the two finite systems just constructed, with their actual cap parameters, factor diagrams and oriented component labels, satisfy exactly the decomposition hypotheses of [F5].

Apply the comparison part of [F5] to those two systems in $L$. It provides the corresponding capped prime factors and actual oriented factor isotopies needed by its lift, including all refinements and suppressed trivial connectors. The correspondence transports the fixed global component names and attachment data from step 1.1.

For each corresponding pair of prime factor diagrams needed in that comparison, apply the imported theorem [F3] to the orientation-preserving pair map induced by its actual capped-factor isotopy. When an intermediate topological planar map $\varphi :S^2\to S^2$ carries one decorated diagram to the next, it gives a finite graph isomorphism preserving the cyclic order at every crossing, the over-under data, and the oriented strand directions. Realize that same graph isomorphism smoothly as follows. Take disjoint crossing disks and thin edge strips for the two connected projection graphs. On each crossing disk choose an orientation-preserving smooth map carrying the four branch germs to their matched germs; on each edge strip choose a smooth monotone arc map that agrees with these maps on smaller endpoint collars. The connected graph's regular neighborhood has disk complements: thicken a spanning tree to a disk, then add the other edge strips one at a time. Each new edge lies in one old disk face and cuts it along a proper arc; the two rounded boundary curves are smooth Jordan curves whose disk sides, by [F7], are exactly the two new complementary disks. By [F7] extend the matched boundary maps over these disk faces with product collars. Gluing produces an orientation-preserving smooth sphere diffeomorphism $g$ inducing exactly $\varphi$'s decorated graph map and component bijection. By [F8], $g$ is joined to the identity by a jointly smooth orientation-preserving isotopy. Thus each topological planar map supplied by [F3] is realized by a sphere planar isotopy in the definition, with the prescribed oriented component correspondence.

The lift in [F5] now applies with its exact marked prime input supplied by step 6.1. It realizes the attachment transfers and prime sequences by finitely many allowed flypes and sphere planar isotopies, giving $D_0$ to $D_1$ with the fixed oriented component correspondence. Together with the crossing-free case, this proves the stated theorem for connected reduced alternating diagrams, including composite links.