  We elaborate on some general aspects of the crossing symmetric approach of
Polyakov to the conformal bootstrap, as recently formulated in Mellin space.
This approach uses, as building blocks, Witten diagrams in AdS. We show the
necessity for having contact Witten diagrams, in addition to the exchange ones,
in two different contexts: a) the large $c$ expansion of the holographic
bootstrap b) in the $\epsilon$ expansion at subleading orders to the ones
studied already. In doing so, we use alternate simplified representations of
the Witten diagrams in Mellin space. This enables us to also obtain compact,
explicit expressions (in terms of a ${}_7F_6$ hypergeometric function!) for the
analogue of the crossing kernel for Witten diagrams i.e., the decomposition
into $s$-channel partial waves of crossed channel exchange diagrams.
