  The puncture method specifies black hole data on a hypersurface with the aid
of a conformal rescaling of the metric that exhibits a coordinate singularity
at the puncture point. When constructing puncture initial data by solving the
Hamiltonian constraint for the conformal factor, the coordinate singularity
requires special attention. The standard way to treat the pole singularity
occurring in wormhole puncture data is not generally applicable to trumpet
puncture data. We investigate a new approach based on inverse powers of the
conformal factor and present numerical examples for single punctures of the
wormhole and 1+log-trumpet type. Additionally, we describe a method to solve
the Hamiltonian constraint for two 1+log trumpets for a given extrinsic
curvature with non-vanishing trace. We investigate properties of this
constructed initial data during binary black hole evolutions and find that the
initial gauge dynamics is reduced.
