  We study the tidal deformations of the shape of a spinning black hole horizon
due to a binary companion in the Bowen-York initial data set. We use the
framework of quasi-local horizons and identify a black hole by marginally outer
trapped surfaces. The intrinsic horizon geometry is specified by a set of mass
and angular-momentum multipole moments $\mathcal{M}_n$ and $\mathcal{J}_n$
respectively.
  The tidal deformations are described by the change in these multipole moments
caused by an external perturbation. This leads us to define two sets of
dimensionless numbers, the tidal coefficients for $\mathcal{M}_n$ and
$\mathcal{J}_n$, which specify the deformations of a black hole with a binary
companion. We compute these tidal coefficients in a specific model problem,
namely the Bowen-York initial data set for binary black holes. We restrict
ourselves to axisymmetric situations and to small spins. Within this
approximation, we analytically compute the conformal factor, the location of
the marginally trapped surfaces, and finally the multipole moments and the
tidal coefficients.
