Essential companion tori for nontrivial companions and incompressible nonproduct patterns

Put $X=e(X_P)$ and $Y=E_K$. They are compact orientable smooth three-manifolds with common boundary torus $T_c$. The exterior $Y$ is connected by [F2]. For completeness $X$ is connected as well: in the standard connected solid torus join two points of $X_P$ by a path consisting of finitely many smooth chart arcs. Remove repeated arcs and perturb finitely many joins in small coordinate balls to obtain an embedded arc, fixed near its endpoints; in dimension three finitely many arcs can be made disjoint except at consecutive endpoints by small finite chart perturbations. By [F5] perturb it to avoid the pattern core $P$. On the punctured supplied tube of $P$, radial projection to its boundary, extended by the identity outside that tube, is continuous and sends this path into $X_P$. Consequently all exterior points are connected by paths. The outer torus has no compressing disk by hypothesis, so [F3], with the entire outer boundary component as subsurface and normal subgroup $1$, gives ${\pi }_1(T_c)\hookrightarrow {\pi }_1(X)$. The companion product tube is exactly the supplied input of [F2], so ${\pi }_1(T_c)\hookrightarrow {\pi }_1(Y)$ too. Both applications are smooth; an abstract finite PL disk has not been treated as a smooth disk.

Use finite smooth collars of the common torus to enlarge the two sides to a connected open cover whose intersection retracts to $T_c$. Van Kampen and the amalgam normal form [F4], with transversals supplied by [A1], then make ${\pi }_1(T_c)$ inject into ${\pi }_1(E_{P(K)})$. Any compressing disk would make its essential torus boundary nullhomotopic in the union, contradicting that injection. Thus $T_c$ is incompressible.

The only boundary of the satellite exterior is the inner pattern torus $T_{\mathrm{i}\mathrm{n}}=\mathrm{\partial }e(N(P))$. The two components cut along $T_c$ are the connected pieces $X$ and $Y$ from step 1.1; only $X$ meets $T_{\mathrm{i}\mathrm{n}}$. Suppose there were a boundary-parallel product region $R$ with ends $T_c,T_{\mathrm{i}\mathrm{n}}$. Its interior must lie in $X$: it cannot cross either of its two boundary tori, and its collar at $T_{\mathrm{i}\mathrm{n}}$ is on the $X$ side. Moreover $R$ exhausts $X$. Indeed $\mathrm{Int}R$ is open in $\mathrm{Int}X$ by the smooth full-dimensional product charts, and closed relative to $\mathrm{Int}X$ because $R$ is compact and its boundary lies in $\mathrm{\partial }X$. The interior of the connected manifold $X$ is connected, as is seen by pushing any boundary portions of a path into its finite collar. Hence $\mathrm{Int}R=\mathrm{Int}X$, and taking closures gives $R=X$. Transporting that product by $e^{-1}$ would identify $X_P$ with a product between its two marked ends, contrary to hypothesis. Therefore $T_c$ is not boundary-parallel. For this closed torus boundary-incompressibility is vacuous, so it is essential by [F6].