Actual same-copy exchange caps have rigid original edge counts

For a proper disk exchange, cut $D$ along its two traces into the outer caps $E,E^{\mathrm{\prime }}$ and middle disk $K$. The irregular resolution of the connected inverse diagram in [F1] joins $E$ to $E^{\mathrm{\prime }}$ along the exchange, producing an embedded proper disk $W$; the remaining sides of $K$ join to form the other component. These pairings are the upper-upper and lower-lower pairs in the transverse four-ray description of [F1]. All other crossings have already been resolved, so these images are embedded. If $\mathrm{\partial }W$ were essential, [F2] would contradict the least weight of $D$. Thus $\mathrm{\partial }W$ is null-homotopic in $\mathrm{\partial }M$.

Form the two cap replacements $A,A^{\mathrm{\prime }}$. For an annular exchange these are $D$ with $E^{\mathrm{\prime }}$ replaced by the disk consisting of the exchange annulus capped by a nearby copy of $E$, and the reverse replacement. They are embedded proper PL disks with boundary exactly $\mathrm{\partial }D$. For a proper disk exchange attach two nearby copies of $E$ to the two frontier arcs of $K$, using the exchange strips, to obtain $A$; use two copies of $E^{\mathrm{\prime }}$ to obtain $A^{\mathrm{\prime }}$. These are again embedded disks. In detail take disjoint small product collars of the finitely many PL cap and strip cells; on each transverse exchange rectangle the two cap copies and the middle rays are joined in separate parallel levels. The maps agree along face boundaries by the supplied exchange neighborhood, and outside that neighborhood the disjoint cut pieces are unchanged. A disk joined to another disk along a boundary interval remains a disk, so the two successive attachments give the claimed disk topology. The collars can be chosen small enough to avoid all remaining surface pieces and to move edge points only along their own original edges.

The proper-disk replacements also have essential boundary. To see this precisely, let $u$ and $v$ be the original-boundary arcs of $E,E^{\mathrm{\prime }}$, transporting the endpoints of $v$ to those of $u$ by the two boundary sides of the exchange disk in $\mathrm{\partial }M$. The circle $\mathrm{\partial }W$ is then $u{v}^{-1}$ with those connectors. Its null homotopy from step 1.1 says that $u$ and the transported $v$ are homotopic with fixed endpoints. The boundary of $A$ is obtained from $\mathrm{\partial }D$ by substituting a transported copy of $u$ for $v$, and the boundary of $A^{\mathrm{\prime }}$ by the reverse substitution. Both are therefore homotopic to $\mathrm{\partial }D$ in $\mathrm{\partial }M$ and are essential. Together with the unchanged boundary in the annular case, both replacements in step 1.2 are eligible essential PL disks.

Let $K$ now denote the remainder of $D$ after removing the two cap interiors; in the annular case it is a disk with two holes, and in the proper case the middle disk. The exchange strips miss every original edge. The collars and copies of step 1.2 consequently give the exact componentwise identities $N(A)=N(K)+2N(E)$ and $N(A^{\mathrm{\prime }})=N(K)+2N(E^{\mathrm{\prime }})$, whereas $N(D)=N(K)+N(E)+N(E^{\mathrm{\prime }})$. In particular $N(A)+N(A^{\mathrm{\prime }})=2N(D)$. Normalize the two essential disks by [F4] to $U,V$. Their counts satisfy $N(U)+N(V)\le 2N(D)$ componentwise. Each has weight at least $w(D)$, while the sum of their weights is at most $2w(D)$. Thus both weights equal $w(D)$ and every componentwise deficit is zero: $N(U)+N(V)=2N(D)$.

Apply the two-output rigidity and lexicographic assertion in [F4]. It gives $v(U)=v(V)=v(D)$ and hence $N(U)=N(V)=N(D)$. Since $N(U)\le N(A)$ and $N(V)\le N(A^{\mathrm{\prime }})$, while their sums are equal by step 3.1, no edge count fell in either normalization. Therefore $N(A)=N(A^{\mathrm{\prime }})=N(D)$. Subtracting their identities in step 3.1 gives $2N(E)=2N(E^{\mathrm{\prime }})$, so $N(E)=N(E^{\mathrm{\prime }})$. Finally [F3] gives their positive total edge weight. All assertions, including the null-homotopic boundary of $W$, now follow. The equality of these edge counts has not been used to infer a relative normal isotopy of the caps.