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We obtain $D_k$ ALF gravitational instantons by a gluing construction whichcaptures, in a precise and explicit fashion, their interpretation as non-linearsuperpositions of the moduli space of centred $SU(2)$ monopoles, equipped withthe Atiyah-Hitchin metric, and $k$ copies of the Taub-NUT manifold. Theconstruction proceeds from a finite set of points in euclidean space,reflection symmetric about the origin, and depends on an adiabatic parameterwhich is incorporated into the geometry as a fifth dimension. Using aformulation in terms of hyperK\"ahler triples on manifolds with boundaries, weshow that the constituent Atiyah-Hitchin and Taub-NUT geometries arise asboundary components of the 5-dimensional geometry as the adiabatic parameter istaken to zero.

$D_k$ Gravitational Instantons as Superpositions of Atiyah-Hitchin and Taub-NUT Geometries