= Nontrivial knot exteriors have incompressible boundary
A <sphere> in a <knot> exterior bounds a ball on the side not containing the <knot>, so the exterior is irreducible. A compression disk for its boundary <torus> would cut the exterior to a manifold with <sphere> boundary; irreducibility makes that manifold a ball, and reversing the compression gives a solid <torus> exterior. Its meridian disk has boundary a preferred <knot> longitude, so adjoining the boundary annulus gives a spanning disk for the <knot>, which is then an <unknot>. The <Loop theorem> therefore makes the peripheral fundamental <group> injective for every nontrivial <knot>. In particular its <knot> <group> contains an injected copy of the <torus> <group> and cannot be cyclic.
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